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href="/categories/%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%BB%9F%E8%AE%A1/%E5%8F%98%E5%88%86%E6%8E%A8%E6%96%AD/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 变分推断</span></a></li><li><a class="site-page child" href="/categories/%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%BB%9F%E8%AE%A1/%E8%BF%91%E4%BC%BC%E8%B4%9D%E5%8F%B6%E6%96%AF%E8%AE%A1%E7%AE%97/"><i class="fa-fw fa-solid fa-cube"></i><span> 近似贝叶斯计算</span></a></li><li><a class="site-page child" href="/categories/%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%BB%9F%E8%AE%A1/%E8%B4%9D%E5%8F%B6%E6%96%AF%E6%A8%A1%E5%9E%8B%E6%AF%94%E8%BE%83%E4%B8%8E%E9%80%89%E6%8B%A9/"><i class="fa-fw fa-solid fa-ghost"></i><span> 模型比较与选择</span></a></li><li><a class="site-page child" href="/categories/%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%BB%9F%E8%AE%A1/%E8%B4%9D%E5%8F%B6%E6%96%AF%E4%BC%98%E5%8C%96/"><i class="fa-fw fa-solid fa-gas-pump"></i><span> 贝叶斯优化</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas 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class="site-page child" href="/categories/BayesNN/%E5%AF%B9%E6%AF%94%E4%B8%8E%E8%AF%84%E6%B5%8B/"><i class="fa-fw fa-brands fa-deezer"></i><span> 对比与评测</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-map"></i><span> 空间统计</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/GeoAI/%E7%BB%BC%E8%BF%B0%E7%B1%BB/"><i class="fa-fw fa-solid fa-hands-holding"></i><span> 概览</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%82%B9%E5%8F%82%E8%80%83%E6%95%B0%E6%8D%AE/"><i class="fa-fw fa-solid fa-map"></i><span> 点参考数据</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E9%9D%A2%E5%85%83%E6%95%B0%E6%8D%AE/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 面元数据</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%82%B9%E6%A8%A1%E5%BC%8F%E6%95%B0%E6%8D%AE/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 点模式数据</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%A9%BA%E9%97%B4%E8%B4%9D%E5%8F%B6%E6%96%AF%E6%96%B9%E6%B3%95/"><i class="fa-fw fa-solid fa-cube"></i><span> 空间贝叶斯方法</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%A9%BA%E9%97%B4%E5%8F%98%E7%B3%BB%E6%95%B0%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-solid fa-ghost"></i><span> 空间变系数模型</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%A9%BA%E9%97%B4%E7%BB%9F%E8%AE%A1%E6%B7%B1%E5%BA%A6%E5%AD%A6%E4%B9%A0/"><i class="fa-fw fa-brands fa-deezer"></i><span> 空间统计深度学习</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E6%97%B6%E7%A9%BA%E7%BB%9F%E8%AE%A1%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fas fa-atlas"></i><span> 时空统计模型</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E5%A4%A7%E6%95%B0%E6%8D%AE%E4%B8%93%E9%A2%98/"><i class="fa-fw fa fa-anchor"></i><span> 大数据专题</span></a></li><li><a class="site-page child" href="/categories/GeoAI/GeoAI/"><i class="fa-fw fa-brands fa-codepen"></i><span> GeoAI</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-database"></i><span> 基础</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E9%AB%98%E7%AD%89%E6%95%B0%E5%AD%A6/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 高等数学</span></a></li><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E6%A6%82%E7%8E%87%E4%B8%8E%E7%BB%9F%E8%AE%A1/"><i class="fa-fw fa-brands fa-deezer"></i><span> 概率与统计</span></a></li><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E7%BA%BF%E4%BB%A3%E4%B8%8E%E7%9F%A9%E9%98%B5%E8%AE%BA/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 线代与矩阵论</span></a></li><li><a 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href="https://xishansnow.github.io/ElementsOfStatisticalLearning/index.html"><i class="fa-fw fa-solid  fa-book-atlas"></i><span> 《统计学习精要（ESL）》</span></a></li><li><a class="site-page child" href="https://xishansnow.github.io/spatialSTAT_CN/index.html"><i class="fa-fw fa-solid  fa-layer-group"></i><span> 《空间统计学》</span></a></li><li><a class="site-page child" target="_blank" rel="noopener" href="https://otexts.com/fppcn/index.html"><i class="fa-fw fa-solid  fa-cloud-sun-rain"></i><span> 《预测：方法与实践》</span></a></li><li><a class="site-page child" href="https://xishansnow.github.io/MLAPP/index.html"><i class="fa-fw fa-solid  fa-robot"></i><span> 《机器学习的概率视角（MLAPP）》</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-compass"></i><span> 索引</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/archives/"><i class="fa-fw fa-solid fa-timeline"></i><span> 时间索引</span></a></li><li><a class="site-page child" href="/tags/"><i class="fa-fw fas fa-tags"></i><span> 标签索引</span></a></li><li><a class="site-page child" href="/categories/"><i class="fa-fw fas fa-folder-open"></i><span> 分类索引</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-link"></i><span> 其他</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/link/food/"><i class="fa-fw fas fa-utensils"></i><span> 美食博主</span></a></li><li><a class="site-page child" href="/link/photography"><i class="fa-fw fas fa-camera"></i><span> 摄影大神</span></a></li><li><a class="site-page child" href="/link/paper/"><i class="fa-fw fas fa-book-open"></i><span> 学术工具</span></a></li><li><a class="site-page child" href="/gallery/"><i class="fa-fw fas fa-images"></i><span> 摄影作品</span></a></li><li><a class="site-page child" href="/about/"><i class="fa-fw fas fa-heart"></i><span> 关于</span></a></li></ul></div></div></div></div><div class="post" id="body-wrap"><header class="post-bg" id="page-header" style="background-image: url('/img/010.png')"><nav id="nav"><span id="blog_name"><a id="site-name" href="/">西山晴雪的知识笔记</a></span><div id="menus"><div id="search-button"><a class="site-page social-icon search"><i class="fas fa-search fa-fw"></i><span> 搜索</span></a></div><div class="menus_items"><div class="menus_item"><a class="site-page" href="/"><i class="fa-fw fas fa-home"></i><span> 主页</span></a></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-atom"></i><span> 预测</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E6%A6%82%E8%A7%88/"><i class="fa-fw fa-solid fa-hands-holding"></i><span> 概览</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E5%B9%BF%E4%B9%89%E7%BA%BF%E6%80%A7%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fas fa-atom"></i><span> 广义线性模型</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E9%9D%9E%E5%8F%82%E6%95%B0%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fas fa-cogs"></i><span> 传统非参数模型</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E9%AB%98%E6%96%AF%E8%BF%87%E7%A8%8B/"><i class="fa-fw fas fa-school"></i><span> 高斯过程</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C/"><i class="fa-fw fas fa-layer-group"></i><span> 神经网络</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E6%A8%A1%E5%9E%8B%E9%80%89%E6%8B%A9%E4%B8%8E%E5%B9%B3%E5%9D%87/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 模型选择与平均</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E5%B0%8F%E6%A0%B7%E6%9C%AC%E5%AD%A6%E4%B9%A0/"><i class="fa-fw fa-solid fa-globe"></i><span> 小样本学习</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-file-export"></i><span> 生成</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E6%A6%82%E8%A7%88/"><i class="fa-fw fa-solid fa-hands-holding"></i><span> 概览</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E4%BC%A0%E7%BB%9F%E6%A6%82%E7%8E%87%E5%9B%BE%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 传统概率图模型</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E6%9C%BA/"><i class="fa-fw fa-solid fa-deezer"></i><span> 玻耳兹曼机</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E5%8F%98%E5%88%86%E8%87%AA%E7%BC%96%E7%A0%81%E5%99%A8/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 变分自编码器</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E8%87%AA%E5%9B%9E%E5%BD%92%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-brands fa-codepen"></i><span> 自回归模型</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E5%BD%92%E4%B8%80%E5%8C%96%E6%B5%81/"><i class="fa-fw fa-solid fa-cube"></i><span> 归一化流</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E6%89%A9%E6%95%A3%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-solid fa-ghost"></i><span> 扩散模型</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E8%83%BD%E9%87%8F%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-solid fa-gas-pump"></i><span> 能量模型</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E7%94%9F%E6%88%90%E5%BC%8F%E5%AF%B9%E6%8A%97%E7%BD%91%E7%BB%9C/"><i class="fa-fw fa-solid fa-globe"></i><span> 生成式对抗网络</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-magnet"></i><span> 挖掘</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E6%A6%82%E8%A7%88/"><i class="fa-fw fa-solid fa-hands-holding"></i><span> 概览</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E9%9A%90%E5%9B%A0%E5%AD%90%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 隐因子模型</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E7%8A%B6%E6%80%81%E7%A9%BA%E9%97%B4%E6%A8%A1%E5%9E%8B/"><i class="fa-fw 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10:42:14">2023-02-09</time></span><span class="post-meta-categories"><span class="post-meta-separator">|</span><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/categories/GeoAI/">GeoAI</a><i class="fas fa-angle-right post-meta-separator"></i><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/categories/GeoAI/%E5%A4%A7%E6%95%B0%E6%8D%AE%E4%B8%93%E9%A2%98/">大数据专题</a><i class="fas fa-angle-right post-meta-separator"></i><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/categories/GeoAI/%E7%82%B9%E5%8F%82%E8%80%83%E6%95%B0%E6%8D%AE/">点参考数据</a></span></div><div class="meta-secondline"><span class="post-meta-separator">|</span><span class="post-meta-wordcount"><i class="far fa-file-word fa-fw post-meta-icon"></i><span class="post-meta-label">字数总计:</span><span class="word-count">26.6k</span><span class="post-meta-separator">|</span><i class="far fa-clock fa-fw 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<link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/hint.css/2.4.1/hint.min.css"><p>【摘 要】 高斯过程和随机场有着悠久的历史，包含了表示空间和时空相关结构的很多方法，例如：协方差函数、谱表示、再生核希尔伯特空间、基于图的模型等。本文介绍了随机偏微分方程方法（SPDE）如何通过 Hilbert 空间投影，将 Matern 协方差模型与其中几种方法建立起联系，并且每种联系在不同情况下都非常有用。除了主要思想的概述之外，本文还讨论了一些重要的扩展、理论、应用和其他新发展。这些方法包括：马尔可夫模型、非马尔可夫模型、非高斯随机场、非平稳场、任意流形上的时空场等，以及实际计算需要考虑的因素。</p>
<p>【原 文】 Lindgren, F., Bolin, D. and Rue, H. (2022) ‘The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running’, Spatial Statistics, 50, p. 100599. Available at: <a target="_blank" rel="noopener" href="https://doi.org/10.1016/j.spasta.2022.100599">https://doi.org/10.1016/j.spasta.2022.100599</a>.</p>
<h2 id="1-简介">1 简介</h2>
<p>关于高斯场的随机偏微分方程 (SPDE) 方法的论文（Lindgren 等，2011 年<sup class="refplus-num"><a href="#ref-Lindgren2011">[91]</a></sup>）发表已经有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span> 年了。本文将借这个机会，从表征高斯随机场分布的不同角度来看待该方法，回顾最近的发展，并展示该方法的一些主要应用。本文还将讨论基于 SPDE 方法的非高斯场构造，该方法最初是由 David Bolin 的博士论文中提出的（Bolin，2012 <sup class="refplus-num"><a href="#ref-Bolin2012">[22]</a></sup>），此后从 Bolin (2014 <sup class="refplus-num"><a href="#ref-Bolin2014">[24]</a></sup>) 开始又发表了一系列论文。</p>
<p>本文以下大部分讨论将集中在 SPDE 及其有限维（希尔伯特空间）表示的性质上，但记住该方法与直接实际相关性也非常重要。本研究的一个重要目标是构建具有良好计算特性的模型，以便人们可以实际使用它们。事实证明，由于稀疏精度矩阵形式及其高效的数值计算，这确实是可能的（Rue 和 Held，2005 年 <sup class="refplus-num"><a href="#ref-Rue2005">[119]</a></sup>；Rue 等，2009 年 <sup class="refplus-num"><a href="#ref-Rue2009">[122]</a></sup>；Martins 等，2013 年 <sup class="refplus-num"><a href="#ref-Martins2013">[96]</a></sup>；Rue 等，2017 年 <sup class="refplus-num"><a href="#ref-Rue2017">[123]</a></sup>；van Niekerk 等，2021 年 <sup class="refplus-num"><a href="#ref-vanNiekerk2021">[148]</a></sup>；Rue 和 Martino，2007 年 <sup class="refplus-num"><a href="#ref-Rue2007">[121]</a></sup>；Eidsvik 等，2009 年 <sup class="refplus-num"><a href="#ref-Eidsvik2009">[50]</a></sup>）。 <code>R-INLA</code>、<code>inlabru</code> 和 <code>rSPDE</code> 包（Krainski 等，2018 年 <sup class="refplus-num"><a href="#ref-Krainski2018">[80]</a></sup>；Bakka 等，2018 年 <sup class="refplus-num"><a href="#ref-Bakka2018">[8]</a></sup>；Lindgren 和 Rue，2015 年 <sup class="refplus-num"><a href="#ref-Lindgren2015">[90]</a></sup>；Bachl 等，2019 年 <sup class="refplus-num"><a href="#ref-Bachl2019">[5]</a></sup>；Bolin 和 Kirchner，2020 年 <sup class="refplus-num"><a href="#ref-Bolin2020K">[28]</a></sup>）为本文的许多讨论提供了基于高斯 SPDE 的模型，而 <code>ngme</code> 包（Asar 等，2020 年 <sup class="refplus-num"><a href="#ref-Asar2020">[2]</a></sup>）实现了非高斯模型。</p>
<h3 id="1-1-协方差矩阵还是精度矩阵？">1.1 协方差矩阵还是精度矩阵？</h3>
<p>SPDE 方法的良好计算特性来自于对精度矩阵的稀疏性考虑，而不是传统的的协方差矩阵，更多细节将在 <code>第 3 节</code> 中出现。</p>
<p>零均值高斯随机向量传统上可以由协方差矩阵表示，其边缘属性可以直接指定或从协方差矩阵中读取，但条件属性必须计算。该随机向量也可以用其精度矩阵（协方差矩阵的逆）表示，这是一种与概率图模型相关的 “现代” 表示法（Lauritzen，1996 <sup class="refplus-num"><a href="#ref-Lauritzen1996">[83]</a></sup>）。在精度矩阵表示中，可以直接指定条件属性或立即读取条件属性，但必须通过计算获得其边缘属性。</p>
<p>在 SPDE 框架内，我们可以将精度矩阵与 “模型的生成方式” 相关联，而将协方差矩阵与 “模型的性质” 相关联。</p>
<p>作为一个简单例子，让我们考虑一个平稳的一阶自回归过程 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>t</mi></msub><mo>=</mo><mi>ϕ</mi><msub><mi>x</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>ϵ</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">x_t = \phi x_{t−1} + \epsilon_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord mathnormal">ϕ</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">ϵ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn><mtext>，</mtext><mo>…</mo><mo separator="true">,</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">t = 1，\ldots ,T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mord cjk_fallback">，</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> 。</p>
<ul>
<li>该过程的精度矩阵是三对角矩阵，因为生成 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">x_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 只需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_{t−1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span></li>
<li>该过程的相关矩阵却是稠密的，其第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mi>j</mi></mrow><annotation encoding="application/x-tex">ij</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">ij</span></span></span></span> 个元素为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>ϕ</mi><mrow><mi mathvariant="normal">∣</mi><mi>i</mi><mo>−</mo><mi>j</mi><mi mathvariant="normal">∣</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\phi^{|i− j|}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0824em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∣</span><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mord mtight">∣</span></span></span></span></span></span></span></span></span></span></span></span>，因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">x_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 依赖于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_{t−1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span>，而 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_{t−1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 又依赖于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>t</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_{t-2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 以此类推。</li>
<li>对于连续时间过程 (Simpson 等, 2012b <sup class="refplus-num"><a href="#ref-Simpson2012b">[135]</a></sup>) 和 Ornstein–Uhlenbeck 过程（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><msub><mi>x</mi><mi>t</mi></msub><mo>=</mo><mo>−</mo><mi>ϕ</mi><msub><mi>x</mi><mi>t</mi></msub><mi>d</mi><mi>t</mi><mo>+</mo><mi>d</mi><msub><mi>B</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">d x_t = −\phi x_t d t + dB_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord mathnormal">ϕ</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">B_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 表示维纳过程），上述密集/稀疏性质依然成立。</li>
</ul>
<p>对于上述一阶自回归过程，使用三对角的精度矩阵进行推断的计算成本为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mclose">)</span></span></span></span>，而基于协方差矩阵的计算成本为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mi>T</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(T^3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。也正式因为因此，在条件独立观测条件下，保持精度矩阵的稀疏形式是卡尔曼递归/更新的关键，并确保了计算成本在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> 中仍然是线性的（Knorr-Held 和 Rue，2002 年<sup class="refplus-num"><a href="#ref-Knorr-Held2002">[79]</a></sup> 的附录）。</p>
<p>如果我们可以同时使用协方差矩阵和精度矩阵这两种方法，那无疑是非常有好处的，因为：<strong>密集的协方差矩阵对于理解单变量的边缘性质和双变量的相关性质非常有用，稀疏的精度矩阵则提供高效的计算能力</strong>。 但由于连续空间中的马尔可夫性质比连续时间中的马尔可夫性质涉及更多因素，因此也变得更为复杂 (Simpson 等, 2012b <sup class="refplus-num"><a href="#ref-Simpson2012b">[135]</a></sup>)。</p>
<p>Lindgren 等 (2011 <sup class="refplus-num"><a href="#ref-Lindgren2011">[91]</a></sup>) 的第一个主要结果表明：通过使用特定函数表征的有限维 Hilbert 空间并将连续域函数投影到该空间，我们可以为具有 Matern 协方差函数的高斯场同时使用精度矩阵和协方差矩阵两种方式。</p>
<h3 id="1-2-最近的一些应用">1.2 最近的一些应用</h3>
<p>我们在此提供一份不完整的 SPDE 方法近期应用列表：</p>
<ul>
<li>天文学（Levis 等，2021 年 <sup class="refplus-num"><a href="#ref-Levis2021">[86]</a></sup>）</li>
<li>健康（Mannseth 等，2021 年 <sup class="refplus-num"><a href="#ref-Mannseth2021">[94]</a></sup>；Scott，2021 年 <sup class="refplus-num"><a href="#ref-Scott2021">[130]</a></sup>；Moses 等 ，2021 年 <sup class="refplus-num"><a href="#ref-Moses2021">[106]</a></sup>；Bertozzi-Villa 等，2021 年 <sup class="refplus-num"><a href="#ref-Bertozzi-Villa2021">[14]</a></sup>；Moraga 等，2021 年 <sup class="refplus-num"><a href="#ref-Moraga2021">[104]</a></sup>；Asri 和 Benamirouche，2021 年<sup class="refplus-num"><a href="#ref-Asri2021">[3]</a></sup>）</li>
<li>工程（Zhang 等，2021 年 <sup class="refplus-num"><a href="#ref-Zhang2021">[167]</a></sup>）</li>
<li>理论（Ghattas 和 Willcox，2021 年<sup class="refplus-num"><a href="#ref-Ghattas2021">[62]</a></sup>；Sanz-Alonso 和 Yang， 2021a <sup class="refplus-num"><a href="#ref-Sanz-Alonso2021a">[127]</a></sup>；Lang 和 Pereira，2021 年<sup class="refplus-num"><a href="#ref-Lang2021">[81]</a></sup>；Bolin 和 Wallin，2021 年<sup class="refplus-num"><a href="#ref-Bolin2021W">[31]</a></sup>）</li>
<li>环境计量学（Roksvåg 等，2021 年 <sup class="refplus-num"><a href="#ref-Roksvag2021a">[116]</a></sup>；Roksvåg 等，2021 年 <sup class="refplus-num"><a href="#ref-Roksvag2021b">[117]</a></sup>；Beloconi 等，2021 年<sup class="refplus-num"><a href="#ref-Beloconi2021">[13]</a></sup>；Vandeskog 等，2021a 年<sup class="refplus-num"><a href="#ref-Vandeskog2021a">[150]</a></sup>；Wang 和 Zuo，2021 年<sup class="refplus-num"><a href="#ref-Wang2021">[157]</a></sup>；Wright 等，2021 年<sup class="refplus-num"><a href="#ref-Wright2021">[162]</a></sup>；Gomez-Catas us 等，2021 年<sup class="refplus-num"><a href="#ref-Gomez-Catas2021">[64]</a></sup>；Valente 和 Laurini，2021b <sup class="refplus-num"><a href="#ref-Valente2021b">[147]</a></sup>；Bleuel 等，2021 年 <sup class="refplus-num"><a href="#ref-Bleuel2021">[20]</a></sup>；Florencio 等，2021 年 <sup class="refplus-num"><a href="#ref-Florencio2021">[56]</a></sup>；Valente 和 Laurini，2021a <sup class="refplus-num"><a href="#ref-Valente2021a">[146]</a></sup>；Hough 等, 2021 <sup class="refplus-num"><a href="#ref-Hough2021">[71]</a></sup>)</li>
<li>计量经济学 (Morales and Laurini, 2021 <sup class="refplus-num"><a href="#ref-Morales2021">[105]</a></sup>; Maynou 等, 2021 <sup class="refplus-num"><a href="#ref-Maynou2021">[98]</a></sup>)</li>
<li>农学 (Borges da Silva 等, 2021 <sup class="refplus-num"><a href="#ref-Borges2021">[35]</a></sup>)</li>
<li>生态学 (Martino 等, 2021 <sup class="refplus-num"><a href="#ref-Martino2021">[95]</a></sup>; Sicacha-Parada 等, 2021 <sup class="refplus-num"><a href="#ref-Sicacha-Parada2021">[131]</a></sup>；Williamson 等，2021 <sup class="refplus-num"><a href="#ref-Williamson2021">[161]</a></sup>；Bell 等，2021 <sup class="refplus-num"><a href="#ref-Bell2021">[12]</a></sup>；Humphreys 等,2021 <sup class="refplus-num"><a href="#ref-Humphreys2021">[72]</a></sup>；Xi 等，2021 <sup class="refplus-num"><a href="#ref-Xi2021">[163]</a></sup>；Fecchio 等,2021 <sup class="refplus-num"><a href="#ref-Fecchio2021">[53]</a></sup> ）</li>
<li>城市规划 (Li, 2021 <sup class="refplus-num"><a href="#ref-Li2021">[87]</a></sup>)</li>
<li>成像 (Aquino 等, 2021 <sup class="refplus-num"><a href="#ref-Aquino2021">[1]</a></sup>)</li>
<li>森林火灾建模 (Taylor 等, 2021 <sup class="refplus-num"><a href="#ref-Taylor2021">[144]</a></sup>; Lindenmayer 等, 2021 <sup class="refplus-num"><a href="#ref-Lindenmayer2021">[88]</a></sup>）</li>
<li>渔业（Babyn 等，2021 年 <sup class="refplus-num"><a href="#ref-Babyn2021">[4]</a></sup>；van Woesik 和 Cacciapaglia，2021 年 <sup class="refplus-num"><a href="#ref-vanWoesik2021">[149]</a></sup>；Jarvis 等，2021 年<sup class="refplus-num"><a href="#ref-Jarvis2021">[74]</a></sup>；Cavieres 等，2021 年 <sup class="refplus-num"><a href="#ref-Cavieres2021">[41]</a></sup>；Monnahan 等，2021 年 <sup class="refplus-num"><a href="#ref-Monnahan2021">[103]</a></sup>；Berg 等，2021 年 <sup class="refplus-num"><a href="#ref-Berg2021">[18]</a></sup>；Breivik 等，2021 年 <sup class="refplus-num"><a href="#ref-Breivik2021">[37]</a></sup>；Lee 等，2021 年<sup class="refplus-num"><a href="#ref-Lee2021">[84]</a></sup>；Thorson 等，2021 年 <sup class="refplus-num"><a href="#ref-Thorson2021">[145]</a></sup>；Griffiths 和 Lezama-Ochoa，2021 年 <sup class="refplus-num"><a href="#ref-Griffiths2021">[65]</a></sup>）</li>
<li>障碍处置（Boman 等，2021 年<sup class="refplus-num"><a href="#ref-Boman2021">[32]</a></sup>；Babyn 等，2021 年<sup class="refplus-num"><a href="#ref-Babyn2021">[4]</a></sup>；Martino 等，2021 年<sup class="refplus-num"><a href="#ref-Martino2021">[95]</a></sup>；Vogel 等，2021 年 <sup class="refplus-num"><a href="#ref-Vogel2021">[153]</a></sup>；Cendoya 等，2021 年 <sup class="refplus-num"><a href="#ref-Cendoya2021">[42]</a></sup>）</li>
<li>等等</li>
</ul>
<h3 id="1-3-本文安排">1.3 本文安排</h3>
<p>本文其余部分的计划如下：</p>
<ul>
<li><code>第 2 节</code> 概述了主要思想，包括精度算子、马尔可夫性质、本征随机场、流形上的随机场、非平稳场和有限元方法 (FEM)。</li>
<li><code>第 3 节</code> 介绍了如何使用 SPDE 方法进行统计推断的主要思想。</li>
<li><code>第 4 节</code> 讨论了非高斯场的扩展，以及具有一般平滑指数的非马尔可夫场的扩展，以及空间和时间上可分离和不可分离模型的扩展。</li>
<li><code>第 5 节</code> 讨论基于 SPDE 模型的理论特性和相应的计算方法，这些模型和方法现在已经从非常一般的条件下的理论角度得到了很好的理解。</li>
<li><code>第 6 节</code> 介绍了一些关键应用，包括疟疾建模、EUSTACE 项目、神经影像学、地震学和生态学中的点过程模型。</li>
<li><code>第 7 节</code> 以相关方法的讨论结束。</li>
<li><code>第 8 节</code> 以一般性讨论结束。</li>
</ul>
<h2 id="2-主要思想概述">2 主要思想概述</h2>
<p>Lindgren 等 (2011 <sup class="refplus-num"><a href="#ref-Lindgren2011">[91]</a></sup>) 的最初动机，旨在解决长期存在的问题，即：如何为高斯马尔可夫随机场 (GMRF) 构造精度矩阵，以便生成的模型对空间邻域定义图的几何形态具有不变性。</p>
<p>该方法的关键点是： <strong>构造一个函数表征的有限维希尔伯特空间，并将连续域（如时间、空间等）函数投影到该空间上。Lindgren 等选择了由局部分段线性基函数张成的有限维 Hillbert 空间，并将连续域上的 Matern 场投影到该空间上，其结果是，连续域上 Matern 场的马尔可夫性会导致新空间中基函数权重的马尔可夫性质。在这种情况下， 与基函数相对应的（三角）剖分图确定了马尔可夫邻域的空间结构，而该模型精度算子（在 <code>第 2.2 节</code> 定义）的阶数则确定了邻域直径</strong>。</p>
<div class="note info no-icon flat"><p>更为直接的理解： 将连续域上的 Matern 场投影到一个由局部分段基函数构造的 Hillbert 空间中；如果该 Martern 场具有马尔可夫性，则这种投影会导致基函数权重系数的马尔可夫性。</p>
</div>
<p>该方法通过有限元数值方法将 “Matern 协方差模型和 SPDE 模型的等价性经典结果”（Matern，1960 年<sup class="refplus-num"><a href="#ref-Matern1960">[97]</a></sup>；Whittle，1954 年<sup class="refplus-num"><a href="#ref-Whittle1954">[158]</a></sup>，1963 年<sup class="refplus-num"><a href="#ref-Whittle1963">[159]</a></sup>）与 “高斯马尔可夫随机场理论”（Besag，1974 年<sup class="refplus-num"><a href="#ref-Besag1974">[15]</a></sup>；Besag 和 Kooperberg，1995 年<sup class="refplus-num"><a href="#ref-Besag1995">[16]</a></sup>；Besag 和 Mondal，2005 年 <sup class="refplus-num"><a href="#ref-Besag2005">[17]</a></sup>；Rue 和 Held，2005 年 <sup class="refplus-num"><a href="#ref-Rue2005">[119]</a></sup>）和 “希尔伯特空间投影” 相结合。</p>
<p>本节重点关注我们得到的结果，以及其与不同随机场表示方法之间的联系，有关技术细节的讨论和最近的理论发展，请参见 <code>第 5 节</code>。</p>
<h3 id="2-1-协方差矩阵和随机偏微分方程">2.1 协方差矩阵和随机偏微分方程</h3>
<p>经典的平稳 Matern 协方差族由下式给出</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>ϱ</mi><mi>M</mi></msub><mrow><mo fence="true">(</mo><mi mathvariant="bold">s</mi><mo separator="true">,</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><mo>=</mo><mfrac><msup><mi>σ</mi><mn>2</mn></msup><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>ν</mi><mo stretchy="false">)</mo><msup><mn>2</mn><mrow><mi>ν</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><msup><mrow><mo fence="true">(</mo><mi>κ</mi><mrow><mo fence="true">∥</mo><mi mathvariant="bold">s</mi><mo>−</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">∥</mo></mrow><mo fence="true">)</mo></mrow><mi>ν</mi></msup><msub><mi>K</mi><mi>ν</mi></msub><mrow><mo fence="true">(</mo><mi>κ</mi><mrow><mo fence="true">∥</mo><mi mathvariant="bold">s</mi><mo>−</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">∥</mo></mrow><mo fence="true">)</mo></mrow><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">\varrho_M\left(\mathbf{s}, \mathbf{s}^{\prime}\right)=\frac{\sigma^2}{\Gamma(\nu) 2^{\nu-1}}\left(\kappa\left\|\mathbf{s}-\mathbf{s}^{\prime}\right\|\right)^\nu K_\nu\left(\kappa\left\|\mathbf{s}-\mathbf{s}^{\prime}\right\|\right),
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">ϱ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathbf">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4271em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Γ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mclose">)</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">κ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">∥</span><span class="mord mathbf">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">∥</span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8562em;"><span style="top:-3.2548em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">κ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">∥</span><span class="mord mathbf">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">∥</span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mi>ν</mi></msub></mrow><annotation encoding="application/x-tex">K_ν</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 为第二类修正贝塞尔函数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ν</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">ν &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 为平滑指数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>κ</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">κ &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">κ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 控制空间相关的变程，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">σ^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 为边缘方差。对于具有 Matern 协方差的场 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u(\cdot )</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span> 中的每个点，有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">E</mi><mo stretchy="false">{</mo><mi>u</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathbb{E}\{u(\mathbf{s})\} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbb">E</span><span class="mopen">{</span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)}</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Cov</mi><mo>⁡</mo><mrow><mo fence="true">{</mo><mi>u</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>u</mi><mrow><mo fence="true">(</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><mo fence="true">}</mo></mrow><mo>=</mo><msub><mi>ϱ</mi><mi>M</mi></msub><mrow><mo fence="true">(</mo><mi mathvariant="bold">s</mi><mo separator="true">,</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\operatorname{Cov}\left\{u(\mathbf{s}), u\left(\mathbf{s}^{\prime}\right)\right\}=\varrho_M\left(\mathbf{s}, \mathbf{s}^{\prime}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mop"><span class="mord mathrm" style="margin-right:0.01389em;">Cov</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">{</span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mclose delimcenter" style="top:0em;">}</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">ϱ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathbf">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span>。</p>
<p>高斯随机场中相关性结构的一个非常有用的广义表征是：对于任意 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span>，线性泛函 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>f</mi><mo separator="true">,</mo><mi>u</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle f, u\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mclose">⟩</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>g</mi><mo separator="true">,</mo><mi>u</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle g, u\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mclose">⟩</span></span></span></span> 之间的协方差可以由下式给出：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="script">R</mi><mi>u</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">Cov</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo stretchy="false">⟨</mo><mi>f</mi><mo separator="true">,</mo><mi>u</mi><mo stretchy="false">⟩</mo><mo separator="true">,</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo separator="true">,</mo><mi>u</mi><mo stretchy="false">⟩</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></msub><msub><mo>∫</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></msub><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mi>ϱ</mi><mrow><mo fence="true">(</mo><mi mathvariant="bold">s</mi><mo separator="true">,</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><mi>g</mi><mrow><mo fence="true">(</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><mi mathvariant="normal">d</mi><mi mathvariant="bold">s</mi><mi mathvariant="normal">d</mi><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">\mathcal{R}_u(f, g)=\operatorname{Cov}(\langle f, u\rangle,\langle g, u\rangle)=\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} f(\mathbf{s}) \varrho\left(\mathbf{s}, \mathbf{s}^{\prime}\right) g\left(\mathbf{s}^{\prime}\right) \mathrm{d} \mathbf{s} \mathrm{d} \mathbf{s}^{\prime}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop"><span class="mord mathrm" style="margin-right:0.01389em;">Cov</span></span><span class="mopen">(⟨</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mclose">⟩</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mclose">⟩)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2719em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.3645em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathbb mtight">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.782em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.3645em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathbb mtight">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.782em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mord mathnormal">ϱ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathbf">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathrm">d</span><span class="mord mathbf">s</span><span class="mord mathrm">d</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">R</mi><mi>u</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{R}_u(f, f)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mclose">)</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">R</mi><mi>u</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{R}_u(g, g)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span></span></span></span> 都是有限的；并且由于 <strong>协方差函数</strong> 或 <strong>核</strong> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϱ</mi><mrow><mo fence="true">(</mo><mi mathvariant="bold">s</mi><mo separator="true">,</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\varrho\left(\mathbf{s}, \mathbf{s}^{\prime}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϱ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathbf">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span> 非负定，因此 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">R</mi><mi>u</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathcal{R}_u(f, f) \geq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>。</p>
<p>对于域 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">D</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.02778em;">D</span></span></span></span>，式中的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>f</mi><mo separator="true">,</mo><mi>u</mi><mo stretchy="false">⟩</mo><mo>=</mo><msub><mo>∫</mo><mi mathvariant="script">D</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mi>u</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold">s</mi></mrow><annotation encoding="application/x-tex">\langle f, u\rangle=\int_{\mathcal{D}} f(\mathbf{s}) u(\mathbf{s}) \mathrm{d} \mathbf{s}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1608em;vertical-align:-0.3558em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1225em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.02778em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3558em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathbf">s</span></span></span></span> 是在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi mathvariant="script">D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L_2(\mathcal{D})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathcal" style="margin-right:0.02778em;">D</span><span class="mclose">)</span></span></span></span> （或者当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> 在不同函数空间中时的对偶配对）上的内积。当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span> 是 Dirac delta 泛函时可以得到点计算，不过这种协方差表征可以扩展到不具有逐点意义的通用随机场。</p>
<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">W</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{W}(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.08222em;">W</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span> 为域 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">D</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.02778em;">D</span></span></span></span> 上的高斯白噪声过程，其特点是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">E</mi><mo stretchy="false">(</mo><mo stretchy="false">⟨</mo><mi>f</mi><mo separator="true">,</mo><mi mathvariant="script">W</mi><mo stretchy="false">⟩</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathbb{E}(\langle f, \mathcal{W}\rangle)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbb">E</span><span class="mopen">(⟨</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathcal" style="margin-right:0.08222em;">W</span><span class="mclose">⟩)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 并且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">R</mi><mi>W</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\mathcal{R}_W(f, g)=\langle f, g\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">W</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">⟩</span></span></span></span>。对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">D</mi><mo>=</mo><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}=\mathbb{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span>，过程 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">W</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{W}(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.08222em;">W</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span> 是布朗运动的形式化导数，但此处的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">W</mi></mrow><annotation encoding="application/x-tex">\mathcal{W}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.08222em;">W</span></span></span></span> 不限于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span>，而是可以定义在更一般的流形上。</p>
<p>根据上述线性泛函协方差的定义，具有如下形式的线性空间随机偏微分方程 (SPDE) 可以被用来定义随机场 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u(\cdot )</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span>：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi mathvariant="script">L</mi><mi>u</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="script">W</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(2)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\mathcal{L} u(\cdot)=\mathcal{W}(\cdot) \tag{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathcal">L</span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.08222em;">W</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span><span class="tag"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">2</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>式中的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">L</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal">L</span></span></span></span> 是 <strong>微分算子</strong>，该算子的选择隐式地决定了方程解中的协方差结构。</p>
<p>当在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span> 上选择如下微分算子时，</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>τ</mi><msup><mrow><mo fence="true">(</mo><msup><mi>κ</mi><mn>2</mn></msup><mo>−</mo><mi mathvariant="normal">Δ</mi><mo fence="true">)</mo></mrow><mrow><mi>α</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><mi>u</mi><mo>=</mo><mi mathvariant="script">W</mi></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(3)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\tau\left(\kappa^2-\Delta\right)^{\alpha / 2} u= \mathcal{W} \tag{3} 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.492em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">Δ</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.142em;"><span style="top:-3.317em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mord mtight">/2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.08222em;">W</span></span><span class="tag"><span class="strut" style="height:1.492em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>Whittle (1954) 和 Whittle (1963) 表明，其稳态解是具有 <code>式 (1)</code>  Matern 协方差的随机场，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ν</mi><mo>=</mo><mi>α</mi><mo>−</mo><mi>d</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">\nu=\alpha - d/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mord">/2</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>ν</mi><mo stretchy="false">)</mo><msup><mrow><mo fence="true">{</mo><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>4</mn><mi>π</mi><msup><mo stretchy="false">)</mo><mrow><mi>d</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><msup><mi>κ</mi><mrow><mn>2</mn><mi>ν</mi></mrow></msup><msup><mi>τ</mi><mn>2</mn></msup><mo fence="true">}</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\sigma^2=\Gamma(\nu)\left\{\Gamma(\alpha)(4 \pi)^{d / 2} \kappa^{2 \nu} \tau^2\right\}^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.442em;vertical-align:-0.35em;"></span><span class="mord">Γ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">{</span></span><span class="mord">Γ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mtight">/2</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">}</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.092em;"><span style="top:-3.3409em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>。我们使用术语 Whittle-Matern 场来指代 <code>式 (3)</code> 方程解的通用集合，其中包括 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span> 上的本征平稳场（将在 <code>第 2.3 节</code>中讨论）、具边界条件的子域上的场、以及在更具一般性的流形上的场。</p>
<h3 id="2-2-精度算子和再生核-Hilbert-空间">2.2 精度算子和再生核 Hilbert 空间</h3>
<p><code>式(2)</code> 解的协方差特征，与某个再生核希尔伯特空间 (RKHS) 的内积密切相关，我们将该内积表示为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">Q</mi><mi>u</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{Q}_u(f, g)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span></span></span></span>。下面概述这种联系的主要要素，更多细节在 <code>附录 A</code> 中给出。</p>
<p>令 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="script">L</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">\mathcal{L}^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathcal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">L</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal">L</span></span></span></span> 的伴随，也就是说，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="script">L</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">\mathcal{L}^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathcal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 是一个能使 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><msup><mi mathvariant="script">L</mi><mo>∗</mo></msup><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>f</mi><mo separator="true">,</mo><mi mathvariant="script">L</mi><mi>g</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle \mathcal{L}^* f, g \rangle=\langle f, \mathcal{L} g \rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathcal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathcal">L</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">⟩</span></span></span></span> 的运算符，并且假设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">L</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal">L</span></span></span></span> 可逆。根据白噪声过程的协方差积 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">R</mi><mi mathvariant="script">W</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{R}_{\mathcal{W}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.08222em;">W</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的定义， <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">⟩</mo><mo>=</mo><msub><mi mathvariant="script">R</mi><mi mathvariant="script">W</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi mathvariant="script">R</mi><mrow><mi mathvariant="script">L</mi><mi>u</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi mathvariant="script">R</mi><mi>u</mi></msub><mrow><mo fence="true">(</mo><msup><mi mathvariant="script">L</mi><mo>∗</mo></msup><mi>f</mi><mo separator="true">,</mo><msup><mi mathvariant="script">L</mi><mo>∗</mo></msup><mi>g</mi><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\langle f, g\rangle=\mathcal{R}_{\mathcal{W}}(f, g)=\mathcal{R}_{\mathcal{L} u}(f, g)=\mathcal{R}_u\left(\mathcal{L}^* f, \mathcal{L}^* g\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.08222em;">W</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathcal mtight">L</span><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathcal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathcal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span>，这表明协方差函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϱ</mi><mrow><mo fence="true">(</mo><mi mathvariant="bold">s</mi><mo separator="true">,</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\varrho\left(\mathbf{s}, \mathbf{s}^{\prime}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϱ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathbf">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span> 满足 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">L</mi><mi mathvariant="bold">s</mi></msub><msub><mi mathvariant="script">L</mi><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></msub><mi>ϱ</mi><mrow><mo fence="true">(</mo><mi mathvariant="bold">s</mi><mo separator="true">,</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><mo>=</mo><msub><mi>δ</mi><mi mathvariant="bold">s</mi></msub><mrow><mo fence="true">(</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathcal{L}_{\mathbf{s}} \mathcal{L}_{\mathbf{s}^{\prime}} \varrho\left(\mathbf{s}, \mathbf{s}^{\prime}\right)=\delta_{\mathbf{s}}\left(\mathbf{s}^{\prime}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1611em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathcal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathbf mtight">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">ϱ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathbf">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1611em;"><span style="top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span>。对于任意 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">s</mi><mo>∈</mo><mi mathvariant="script">D</mi></mrow><annotation encoding="application/x-tex">\mathbf{s} \in \mathcal{D}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathbf">s</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.02778em;">D</span></span></span></span> 和合适的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span>，这可以用来证明 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">Q</mi><mi>u</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi mathvariant="script">L</mi><mi>f</mi><mo separator="true">,</mo><mi mathvariant="script">L</mi><mi>g</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\mathcal{Q}_u(f, g)=\langle\mathcal{L} f, \mathcal{L} g\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathcal">L</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathcal">L</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">⟩</span></span></span></span> 满足 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">Q</mi><mi>u</mi></msub><mo stretchy="false">{</mo><mi>ϱ</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">,</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{Q}_u\{\varrho(\mathbf{s}, \cdot), g(\cdot)\}=g(\mathbf{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">{</span><span class="mord mathnormal">ϱ</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">⋅</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)}</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span></span></span></span>, 这意味着 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">Q</mi><mi>u</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo separator="true">,</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{Q}_u(\cdot, \cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">⋅</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span>  是 RKHS 中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϱ</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo separator="true">,</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\varrho(\cdot ,\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϱ</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span> 的内积。</p>
<p>此外，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><msup><mi mathvariant="script">L</mi><mo>∗</mo></msup><mi mathvariant="script">L</mi><mi>ϱ</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">,</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">⟩</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\langle \mathcal{L}^*\mathcal{L} \varrho(\mathbf{s},\cdot ),g\rangle = g(\mathbf{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathcal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord mathcal">L</span><span class="mord mathnormal">ϱ</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">⋅</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span></span></span></span>，这意味着协方差函数是 <strong>精度算子（precision operator）</strong> 的格林函数（此处为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="script">L</mi><mo lspace="0em" rspace="0em">∗</mo></msup><mi mathvariant="script">L</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}^{*}\mathcal{L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathcal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span></span></span></span></span><span class="mord mathcal">L</span></span></span></span>）。因此，对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span> 上的 Whittle-Matern 过程，有：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="script">Q</mi><mi>u</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>τ</mi><mn>2</mn></msup><mo stretchy="false">⟨</mo><mo stretchy="false">(</mo><msup><mi>κ</mi><mn>2</mn></msup><mo>−</mo><mi mathvariant="normal">Δ</mi><msup><mo stretchy="false">)</mo><mrow><mi>α</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><mi>f</mi><mo separator="true">,</mo><mo stretchy="false">(</mo><msup><mi>κ</mi><mn>2</mn></msup><mo>−</mo><mi mathvariant="normal">Δ</mi><msup><mo stretchy="false">)</mo><mrow><mi>α</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><mi>g</mi><mo stretchy="false">⟩</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(4)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\mathcal{Q}_u(f, g) = \tau^2 \langle(\kappa^{2} − \Delta)^{\alpha/2} f, (\kappa^{2} − \Delta)^{\alpha/2} g\rangle \tag{4}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">⟨(</span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mord mtight">/2</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mord mtight">/2</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">⟩</span></span><span class="tag"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">4</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p><strong>精度算子</strong> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>τ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><msup><mi>κ</mi><mn>2</mn></msup><mo>−</mo><mi mathvariant="normal">Δ</mi><msup><mo stretchy="false">)</mo><mi>α</mi></msup></mrow><annotation encoding="application/x-tex">\tau^2 (\kappa^{2} − \Delta)^{\alpha}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span></span></span></span></span></span></span></span>。</p>
<p>对于高斯场，高阶马尔可夫性质可以通过（由局部微分算子定义的）精度算子来表征 (Rozanov, 1977 <sup class="refplus-num"><a href="#ref-Rozanov1977">[118]</a></sup>)。在 Matern 中的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">α</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 具有整数值的情况下，精度算子扩展为 “负拉普拉斯算子的整数幂之和”，表明相应过程是一个马尔可夫随机场。此外，<code>式(4)</code> 中内积本身可以扩展为 “仅涉及拉普拉斯算子整数和半整数幂以及梯度的内积之和”：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi mathvariant="script">Q</mi><mi>u</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi mathvariant="normal">k</mi><mo>=</mo><mn>0</mn></mrow><mi>α</mi></munderover><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mi>α</mi><mi mathvariant="normal">k</mi></mfrac><mo fence="true">)</mo></mrow><msup><mi>κ</mi><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>α</mi><mo>−</mo><mi mathvariant="normal">k</mi><mo stretchy="false">)</mo></mrow></msup><mrow><mo fence="true">⟨</mo><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="normal">Δ</mi><msup><mo stretchy="false">)</mo><mrow><mi mathvariant="normal">k</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><mi>f</mi><mo separator="true">,</mo><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="normal">Δ</mi><msup><mo stretchy="false">)</mo><mrow><mi mathvariant="normal">k</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><mi>g</mi><mo fence="true">⟩</mo></mrow></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(5)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\mathcal{Q}_u(f, g) = \sum^{\alpha}_{\mathrm{k}=0} \binom{\alpha}{\mathrm{k}} \kappa^{2(α−\mathrm{k})} \left\langle(−\Delta)^{\mathrm{k}/2} f,(−\Delta)^{\mathrm{k}/2} g \right \rangle\tag{5} 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9535em;vertical-align:-1.3021em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">k</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">k</span></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mbin mtight">−</span><span class="mord mathrm mtight">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">⟨</span></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">Δ</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">k</span><span class="mord mtight">/2</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">Δ</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">k</span><span class="mord mtight">/2</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">⟩</span></span></span></span><span class="tag"><span class="strut" style="height:2.9535em;vertical-align:-1.3021em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>其中半整数拉普拉斯的内积，可以等效地被转换为梯度算子的内积，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="normal">Δ</mi><msup><mo stretchy="false">)</mo><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><mi>f</mi><mo separator="true">,</mo><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="normal">Δ</mi><msup><mo stretchy="false">)</mo><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><mi>g</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo separator="true">,</mo><mi mathvariant="normal">∇</mi><mi>g</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle(−\Delta)^{1/2} f, (−\Delta)^{1/2} g\rangle = \langle \nabla  f, \nabla g \rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mopen">⟨(</span><span class="mord">−</span><span class="mord">Δ</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1/2</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">Δ</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1/2</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">⟩</span></span></span></span>（Lindgren 等, 2011 <sup class="refplus-num"><a href="#ref-Lindgren2011">[91]</a></sup>, 附录 B)。这表明在内积积分中，Matern 场的马尔可夫子集仅涉及局部微分算子（尽管半拉普拉斯算子是非局部算子）。这在构造精度算子的离散化表示时非常有用；连续域上的全局马尔可夫性质（通过子域的条件独立性以及它们之间的分隔集来表达）被转变成了相似的条件，该性质被传送到（三角）剖分图的高阶邻域拓扑中，并导致精度算子的稀疏矩阵表示（<code>第 2.7 节</code>）。</p>
<p>RKHS 构造、样条和随机过程估计之间的联系由来已久，Kimeldorf 和 Wahba (1970 <sup class="refplus-num"><a href="#ref-Kimeldorf1970">[77]</a></sup>) 展示了如何将高斯过程回归中的条件期望转换为惩罚样条。 RKHS 理论的样条和高斯随机场具有一个重要区别：虽然样条是在紧凑域上具有有限精度范数的 RKHS 的成员，但与该 RKHS 相关的随机场实现并没有有限范数。这样做的原因是：随机场的实现很少是平滑的，只有场的条件期望才是 RKHS 的适当成员。</p>
<h3 id="2-3-唯一性和本征平稳随机场">2.3 唯一性和本征平稳随机场</h3>
<p>精度算子的格林函数不一定是唯一的，例如当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">Q</mi><mi>u</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{Q}_u(f,f)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mclose">)</span></span></span></span> 只生成一个半范数时。为了避免由此产生的额外解空间，基础 Matern 情况需要对平稳解进行限制，以消除该算子零空间中的函数，例如对于任何 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">s</mi><mn>0</mn></msub><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{s}_0 \in  \mathbb{R}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∥</mi><msub><mi mathvariant="bold">s</mi><mn>0</mn></msub><mi mathvariant="normal">∥</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\| \mathbf{s}_0 \| = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 时的函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>κ</mi><mrow></mrow><mo>⋅</mo><msub><mi mathvariant="bold">s</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(κ \mathbf{} \cdot \mathbf{s}_0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord mathnormal">κ</span><span class="mord"></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。对于紧凑域，此类型的场通常受确定性 Neumann 边界条件的限制，导致域边界附近的非平稳行为（<code>第 5.1 1 节</code> 中有更多详细信息）。</p>
<p>当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">Q</mi><mi>u</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo separator="true">,</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{Q}_u(\cdot ,\cdot )</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">⋅</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span> 的零空间未被边界条件或其他条件消除时，通过对场应用某种对比滤波器也能实现平稳性，结果是一系列本征平稳模型，当向对比算子的零空间中添加函数时，模型的概率结构具有不变性。经典的基于网格的本征平稳随机场（Besag 和 Kooperberg，1995 <sup class="refplus-num"><a href="#ref-Besag1995">[16]</a></sup>；Besag 和 Mondal，2005 <sup class="refplus-num"><a href="#ref-Besag2005">[17]</a></sup>）对应于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>κ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">κ = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">κ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 时的连续域模型，这给出了常量（对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">α = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>）和平面（对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">α = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>）相加的不变性。不过，当精度内积被推广到（例如）振荡场模型时，可能会出现更复杂类型的零空间。例如，对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span> 上的振荡场，Lindgren 等（2011 年 <sup class="refplus-num"><a href="#ref-Lindgren2011">[91]</a></sup>，第 3.3 节）针对其无阻尼极限情况，给出了波数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">κ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">κ</span></span></span></span> 时，任意方向正弦和余弦函数的不变性。</p>
<h3 id="2-4-谱表示">2.4 谱表示</h3>
<p>对于平稳模型的理论处理来说，谱表示必不可少。高斯过程的经典线性滤波器理论可以直接应用，我们推荐参阅 Cramer 和 Leadbetter (2004 <sup class="refplus-num"><a href="#ref-Cramer2004">[46]</a></sup>) 以及 Lindgren (2012 <sup class="refplus-num"><a href="#ref-Lindgren2012">[92]</a></sup>) 的理论基础。</p>
<p>平稳高斯过程的协方差函数可以写成（对称非负）谱测度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>S</mi><mo stretchy="false">(</mo><mi mathvariant="bold">k</mi><mo stretchy="false">)</mo><mtext>，</mtext><mi mathvariant="bold">k</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">dS(\mathbf{k})，\mathbf{k} \in \mathbb{R}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathbf">k</span><span class="mclose">)</span><span class="mord cjk_fallback">，</span><span class="mord mathbf">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span> 的傅里叶变换形式：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>ϱ</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">,</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></msub><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>−</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi mathvariant="bold">k</mi><mo stretchy="false">)</mo><mi>d</mi><mi>S</mi><mo stretchy="false">(</mo><mi mathvariant="bold">k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\varrho(\mathbf{s},\mathbf{s}^{\prime}) = \int_{\mathbb{R}^d} \exp(i(\mathbf{s}^{\prime} − \mathbf{s}) \cdot  \mathbf{k}) dS(\mathbf{k})
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϱ</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2719em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.3645em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathbb mtight">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.782em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">k</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathbf">k</span><span class="mclose">)</span></span></span></span></span></p>
<p>当测度允许密度时，我们写成 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>S</mi><mo stretchy="false">(</mo><mi mathvariant="bold">k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><mi mathvariant="bold">k</mi><mo stretchy="false">)</mo><mi>d</mi><mi mathvariant="bold">k</mi></mrow><annotation encoding="application/x-tex">dS(\mathbf{k}) = S (\mathbf{k}) d \mathbf{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathbf">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathbf">k</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathbf">k</span></span></span></span>，过程本身可以被构造为随机傅里叶积分：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></msub><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>i</mi><mi mathvariant="bold">s</mi><mo>⋅</mo><mi mathvariant="bold">k</mi><mo stretchy="false">)</mo><msqrt><mrow><mi>S</mi><mo stretchy="false">(</mo><mi mathvariant="bold">k</mi><mo stretchy="false">)</mo></mrow></msqrt><mi>d</mi><mi>Z</mi><mo stretchy="false">(</mo><mi mathvariant="bold">k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u(\mathbf{s}) = \int_{\mathbb{R}^d} \exp(i \mathbf{s} \cdot  \mathbf{k}) \sqrt{S(\mathbf{k})} d Z(\mathbf{k})
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2719em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.3645em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathbb mtight">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.782em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mord mathbf">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.2561em;"></span><span class="mord mathbf">k</span><span class="mclose">)</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9839em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathbf">k</span><span class="mclose">)</span></span></span><span style="top:-2.9439em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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<p>式中的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>Z</mi><mo stretchy="false">(</mo><mi mathvariant="bold">k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d Z(\mathbf{k})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathbf">k</span><span class="mclose">)</span></span></span></span> 是以复数值为中心的高斯白噪声测度，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>Z</mi><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="bold">k</mi><mo stretchy="false">)</mo><mo>=</mo><mover accent="true"><mrow><mi>d</mi><mi>Z</mi><mo stretchy="false">(</mo><mi mathvariant="bold">k</mi><mo stretchy="false">)</mo></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">d Z(-\mathbf{k}) = \overline{d Z(\mathbf{k})}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathbf">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathbf">k</span><span class="mclose">)</span></span></span><span style="top:-3.87em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.25em;"><span></span></span></span></span></span></span></span></span> ，并且协方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Cov</mi><mo>⁡</mo><mo stretchy="false">{</mo><mi>d</mi><mi>Z</mi><mo stretchy="false">(</mo><mi mathvariant="bold">k</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>d</mi><mi>Z</mi><mo stretchy="false">(</mo><msup><mi mathvariant="bold">k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>=</mo><mi>δ</mi><mo stretchy="false">(</mo><mi mathvariant="bold">k</mi><mo>−</mo><msup><mi mathvariant="bold">k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo><mi>d</mi><mi mathvariant="bold">k</mi></mrow><annotation encoding="application/x-tex">\operatorname{Cov}\{d Z(\mathbf{k}), d Z(\mathbf{k}^{\prime})\} = \delta (\mathbf{k} - \mathbf{k}^{\prime}) d \mathbf{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mop"><span class="mord mathrm" style="margin-right:0.01389em;">Cov</span></span><span class="mopen">{</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathbf">k</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)}</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span><span class="mopen">(</span><span class="mord mathbf">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathbf">k</span></span></span></span>。</p>
<p>SPDE 和 Matern 协方差之间的一般联系由 Whittle (1963 <sup class="refplus-num"><a href="#ref-Whittle1963">[159]</a></sup>) 给出了证明，通过使用微分算子的谱表示，表明 <code>式(1)</code> 的 Matern 协方差是谱密度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S_M(\mathbf{k})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">k</span><span class="mclose">)</span></span></span></span> 的傅里叶变换，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">k</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><msup><mi>τ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo><mi>d</mi></msup><mo stretchy="false">(</mo><msup><mi>κ</mi><mn>2</mn></msup><mo>+</mo><mi mathvariant="normal">∥</mi><mi>k</mi><msup><mi mathvariant="normal">∥</mi><mn>2</mn></msup><msup><mo stretchy="false">)</mo><mi>α</mi></msup><msup><mo stretchy="false">}</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo separator="true">,</mo><mi mathvariant="bold">k</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">S_M(\mathbf{k}) = \{\tau^2 (2\pi)^d(\kappa^{2} + \| k \|^2)^α\}^{−1}, \mathbf{k} \in  \mathbb{R}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">}</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span>，从 <code>式(3)</code> 解的精度算子的谱表示的倒数得到。谱密度中出现的因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">(2\pi)^d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span> 来自标准化白噪声定义的谱密度，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">R</mi><mi>W</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\mathcal{R}_W(f,g)=\langle f,g \rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">W</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">⟩</span></span></span></span>，并且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∥</mi><mi mathvariant="bold">k</mi><msup><mi mathvariant="normal">∥</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\|\mathbf{k}\|^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mord mathbf">k</span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span> 上 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">Δ</span></span></span></span> 的特征值。马尔可夫性质的局部精度算子表征意味着：当且仅当谱密度的倒数为偶数多项式时，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span> 上的平稳过程是马尔可夫的（Rozanov 1977 <sup class="refplus-num"><a href="#ref-Rozanov1977">[118]</a></sup>)。我们可以看到，当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">α</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 为整数时满足该条件。</p>
<h3 id="2-5-流形">2.5 流形</h3>
<p>Lindgren 等 (2011 <sup class="refplus-num"><a href="#ref-Lindgren2011">[91]</a></sup>) 的一个启发性例子是在球面上构建平稳 MRF 模型，以解决历史气候建模等地球科学问题。有关历史联系，请参阅 Wahba (1981 <sup class="refplus-num"><a href="#ref-Wahba1981">[154]</a></sup>)，其中在对球面数据建模时使用了与 <code>式 (5)</code> 内积类似形式的样条惩罚。</p>
<p>与显式的协方差指定相比， <strong>SPDE 方法的优势在于可以轻松地在任何足够平滑的流形上构建范围广泛的有效模型</strong>，Whittle 表征为此类流形的 Matern 场提供了自然泛化。通过令 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> 表示球面上的 Laplace-Beltrami 算子（这是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span> 的拉普拉斯算子在球面上受到的约束），可以直接使用连续域中精度算子的定义，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">D</mi><mo>=</mo><msup><mi mathvariant="double-struck">S</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathcal{D} = \mathbb{S}^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>。有限维 Hilbert 空间表示的基本收敛性证明（参见第 2.7 节）与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span> 的紧凑扁平子域的证明相同。</p>
<p>在处理非欧几里德流形时，谱理论可能不如 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span> 中直观，但对于任何光滑紧致的流形而言，泛化到流形上的拉普拉斯算子（如球面上的 Laplace-Beltrami 算子）的特征函数集合，形成了一个可数的特征函数基，并可被用于构造类似傅里叶的表示。精度算子、协方差函数和过程本身的谱表示，遵循与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span> 相同的原则，但具有可数的调和基。该技术被用于证明广义格林恒等式引理（ Lindgren 等 2011 年 <sup class="refplus-num"><a href="#ref-Lindgren2011">[91]</a></sup>，附录 D）。在球面上，平稳高斯场的球谐表示变为</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><munderover><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mo>−</mo><mi>k</mi></mrow><mi>k</mi></munderover><msub><mi>Y</mi><mrow><mi>k</mi><mo separator="true">,</mo><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><msqrt><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msqrt><msub><mi>Z</mi><mrow><mi>k</mi><mo separator="true">,</mo><mi>m</mi></mrow></msub><mo separator="true">,</mo><mspace width="2em"></mspace><mi mathvariant="bold">s</mi><mo>∈</mo><msup><mi mathvariant="bold">S</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">u(\mathbf{s}) = \sum^{\infty}_{k=0} \sum^{k}_{m=-k} Y_{k,m}(\mathbf{s}) \sqrt{S(k)} Z_{k,m},\qquad \mathbf{s} \in  \mathbf{S}^2
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.1966em;vertical-align:-1.3604em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8361em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mrel mtight">=</span><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3604em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9839em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span><span style="top:-2.9439em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>Y</mi><mrow><mi>k</mi><mo separator="true">,</mo><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>λ</mi><mi>k</mi></msub><msub><mi>Y</mi><mrow><mi>k</mi><mo separator="true">,</mo><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta Y_{k,m}(\cdot ) = λ_kY_{k,m}(\cdot )</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span>，特征值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>k</mi></msub><mo>=</mo><mo>−</mo><mi>k</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mtext>，</mtext><mi>k</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo></mrow><annotation encoding="application/x-tex">\lambda_k = −k(k + 1)，k = 0, 1, 2,\ldots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord cjk_fallback">，</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span></span></span></span>, 内求和次数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">2k + 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>（ <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>=</mo><mo>−</mo><mi>k</mi><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">m = −k,\ldots, k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> ），<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Z</mi><mrow><mi>k</mi><mo separator="true">,</mo><mi>m</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Z_{k,m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 为独立的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}(0, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.14736em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 变量。利用勒让德多项式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_k(\cdot )</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span>，可以用谱表示来计算协方差：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>ϱ</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">,</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>S</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><msub><mi>P</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo>⋅</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="2em"></mspace><mi mathvariant="bold">s</mi><mo separator="true">,</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>∈</mo><msup><mi mathvariant="bold">S</mi><mn>2</mn></msup></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(6)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\varrho(\mathbf{s},\mathbf{s}^{\prime}) = \sum^{\infty}_{k=0} (2k + 1) S(k)P_k(\mathbf{s}\cdot \mathbf{s}^{\prime}), \qquad \mathbf{s}, \mathbf{s}^{\prime} \in  \mathbf{S}^2 \tag{6}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϱ</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9535em;vertical-align:-1.3021em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathbf">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:2.9535em;vertical-align:-1.3021em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">6</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">2k + 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 的因子来自球谐函数的求和/乘积公式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mo>−</mo><mi>k</mi></mrow><mi>k</mi></msubsup><msub><mi>Y</mi><mrow><mi>k</mi><mo separator="true">,</mo><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><msub><mi>Y</mi><mrow><mi>k</mi><mo separator="true">,</mo><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>P</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo>⋅</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sum^{k}_{m=−k} Y_{k,m}(\mathbf{s}) Y_{k,m}(\mathbf{s}^{\prime}) = (2k+1) P_k(\mathbf{s}\cdot \mathbf{s}^{\prime})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.347em;vertical-align:-0.358em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.989em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mrel mtight">=</span><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。有关更多理论背景，请参阅 Schoenberg (1942 <sup class="refplus-num"><a href="#ref-Schoenberg1942">[129]</a></sup>) 和 Wahba (1981 <sup class="refplus-num"><a href="#ref-Wahba1981">[154]</a></sup>)。</p>
<p>将线性滤波器理论应用于球面上的 Whittle SPDE，会产生 Whittle-Matern 谱 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>4</mn><mi>π</mi><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>τ</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo stretchy="false">{</mo><msup><mi>κ</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mo stretchy="false">}</mo><mrow><mo>−</mo><mi>α</mi></mrow></msup><mo separator="true">,</mo><mi>k</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo></mrow><annotation encoding="application/x-tex">S_M(k) = (4\pi)^{−1}τ^{−2} \{\kappa^{2} + k(k + 1)\}^{−α}, k = 0, 1 , 2,\ldots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0213em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mclose"><span class="mclose">}</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span></span></span></span>。协方差矩阵在封闭形式下不可用，但可以通过 <code>式(6)</code> 的无限级数进行数值计算，它对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">α &gt; 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 是收敛的，对应的平滑度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ν</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">ν &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p>
<p>请注意，马尔可夫特征取决于局部精度算子 ( Rozanov 1977 <sup class="refplus-num"><a href="#ref-Rozanov1977">[118]</a></sup>) ，因此即使谱的函数形式的倒数不是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 的偶数多项式，整数的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">α</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 值仍会在球面上生成马尔可夫过程。</p>
<p>域的流形曲率会影响精度算子的格林函数，因此“平稳” 场的概念在很大程度上局限于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">S</mi><mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{S}^d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathbb">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span> 上的场。对于其他流形而言，除了已经存在的具有边界的紧凑域的边界效应之外，其微分几何结构变得非常重要。</p>
<h3 id="2-6-非平稳模型">2.6 非平稳模型</h3>
<p>一旦建立了平稳 Matern 场和随机偏微分方程之间的联系，就可以通过多种方式扩展模型族。除了一般的流形扩展之外，还可以通过修改微分算子来构建非平稳模型。 <strong>广义 Whittle-Matern 模型</strong> 是一个直接的非平稳扩展，该模型让 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">κ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">κ</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">τ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span></span></span> 取决于位置，并将拉普拉斯算子扩展到非平稳各向异性版本：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mo stretchy="false">{</mo><mi>κ</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><mi mathvariant="normal">∇</mi><mo>⋅</mo><mi mathvariant="bold">H</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∇</mi><msup><mo stretchy="false">}</mo><mrow><mi>α</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><mi>u</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>τ</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo></mrow></mfrac><mi mathvariant="script">W</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(7)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\{κ(\mathbf{s})^2 − \nabla \cdot \mathbf{H}(\mathbf{s}) \nabla \}^{α/2} u(\mathbf{s}) = \frac{1}{τ(\mathbf{s})} \mathcal{W}(\mathbf{s}) \tag{7}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">κ</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord mathbf">H</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mord">∇</span><span class="mclose"><span class="mclose">}</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mord mtight">/2</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2574em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathcal" style="margin-right:0.08222em;">W</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span></span><span class="tag"><span class="strut" style="height:2.2574em;vertical-align:-0.936em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">7</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>由于参数场只有温和的正则条件（参见<code>第 5.1 2 节</code>），该模型会产生隐式定义的正定非平稳协方差函数，并且精度算子在封闭形式中可用。 Lindgren 等 (2011) 的温度应用使用此广义模型，其中令 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">α = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">H</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mo>≡</mo><mi mathvariant="bold">I</mi></mrow><annotation encoding="application/x-tex">\mathbf{H}(\mathbf{s}) \equiv \mathbf{I}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">H</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">I</span></span></span></span>，并 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mtext>纬度</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sin(纬度)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">in</span><span class="mopen">(</span><span class="mord cjk_fallback">纬度</span><span class="mclose">)</span></span></span></span> 中的三段二次 B 样条基函数来表示 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">κ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">κ</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">τ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span></span></span> 的对数线性模型。 Ingebrigtsen 等（2014 <sup class="refplus-num"><a href="#ref-Ingebrigtsen2014">[73]</a></sup>）使用了空间地理协变量。Yue 等 (2014 <sup class="refplus-num"><a href="#ref-Yue2014">[165]</a></sup>) 讨论了自适应样条模型的应用，Fuglstad 等 (2015a <sup class="refplus-num"><a href="#ref-Fuglstad2015a">[58]</a></sup>) 探索了实用的各向异性拉普拉斯算子表示。</p>
<p>更一般的非平稳模型的主要挑战在于推断实践，因为在许多情况下，只有单一的噪声场可用，这使得实际可识别性成为一个挑战（参见 Fuglstad 等，2015b <sup class="refplus-num"><a href="#ref-Fuglstad2015b">[59]</a></sup>；Bolin 和 Kirchner，2021 <sup class="refplus-num"><a href="#ref-Bolin2021K">[29]</a></sup>）。但这并不是 SPDE 构造的独有特征，因为任何足够通用的非平稳模型族都是如此。通过利用算子的性质，一种可能的方法是在局部估计算子，避免全局计算。只要强制执行基本的正则条件，结果就会产生一个有效的全局模型，并且可以将更多的精力用于改进局部估计，而不是处理非平稳协方差函数的繁琐的正定性问题。</p>
<p>Bakka 等 (2019 <sup class="refplus-num"><a href="#ref-Bakka2019">[6]</a></sup>) 介绍了这种方法的一个特例。他们以障碍模型的形式，断开跨越物理障碍的过程，阻止虚假依赖在 “欧几里得距离上很近但在测地距离上很远的” 点之间传播。该想法让 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">κ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">κ</span></span></span></span> 在形成障碍的区域接近 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span>（这与空间相关变程接近零具有相同的效果），而在感兴趣域中采用恒定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">κ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">κ</span></span></span></span> 值。与确定性 Neumann 条件相比，由此产生的场几乎没有边界效应，这使其成为许多实际情况的有吸引力的替代方案，例如在岛群中模拟鱼类，此情况下的依赖性不应跨越陆地。</p>
<p>改进 SPDE 算子相当于改变黎曼流形上的计量指标 (Lindgren 等，2011)。这为非平稳随机场的经典变形方法（ Sampson 和 Guttorp, 1992 <sup class="refplus-num"><a href="#ref-Sampson1992">[124]</a></sup> ）提供了一个有启发性的比较。变形方法的工作原理是将目标场的域变形，有选择地将其变形为嵌入更高维度的流形。然后，将平稳协方差模型应用于变形后的流形。当映射回原始空间时，将获得一个非平稳的模型（ Hildeman 等，2021 年 <sup class="refplus-num"><a href="#ref-Hildeman2021">[70]</a></sup>）。相反，如果将 <code>式(3)</code> 的基本 Whittle SPDE 模型应用于变形后的流形域，则会获得一个非常不同的非稳态模型，该模型与流形内的测地线距离相关。通过构造流形的计量指标并将表达式转换回原始流形坐标，可以获得类似于 <code>式（7）</code> 的非平稳 SPDE 模型。这提供了一种参数化某些类型非平稳性的不同方法。这种解释的一个主要好处是：它提供了几何可解释性，既针对 SPDE 模型本身，也针对其与经典变形方法的不同。对于给定流形，这种方法隐含地遵循了非平稳性，但有时很难找到能够生成特定预定义非平稳行为的流形。 Fuglstad 等 (2019 <sup class="refplus-num"><a href="#ref-Fuglstad2019">[60]</a></sup>) 的补充材料 S7.1 中，童工了后者的示例，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>κ</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">κ(\mathbf{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">κ</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span></span></span></span> 的分段线性变化对应于圆柱变形流形。然而，直接设计这种类型的模型具有挑战性，因为嵌入空间可能需要比 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span> 大得多才能捕获所需的性质。但最大的收获在于，非平稳 SPDE 算子和流形上的计量指标之间密切相关。</p>
<h3 id="2-7-局部支持的希尔伯特空间基和离散精度">2.7 局部支持的希尔伯特空间基和离散精度</h3>
<p>为了构建 SPDE 解的有限维表示，Lindgren 等 (2011) 使用分段线性基函数在空间三角剖分上实现了局部支持。这种选择保留了连续马尔可夫性质的许多优点，在以地理参考观测为条件时也会导致稀疏矩阵，这与调和基函数和 Karhunen-Loeve 展开式等其他非局部基选择形成了鲜明对比。希尔伯特空间投影理论对所有这些选择的工作原理基本相同，但我们现在将关注其马尔可夫版本，关于非局部基的选择问题，参见<code>第 7 节</code>。</p>
<p>令 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>ψ</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>N</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\psi_j(\mathbf{s}), j = 1,\ldots , N\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mclose">}</span></span></span></span> 表示每个空间位置 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">s</mi></mrow><annotation encoding="application/x-tex">\mathbf{s}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">s</span></span></span></span> 处的一组连续分段的线性基函数，其总和为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，且每个都支持连接到某个顶点的三角形。对于平面三角剖分，每个顶点的平均三角形数约为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">6</span></span></span></span>。然后我们寻找基权重向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">u</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>u</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>u</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>u</mi><mi>N</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbf{u} = \{u_1, u_2,\ldots , u_N\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>，以使结果函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>u</mi><mo stretchy="true">~</mo></mover><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><msub><mi>ψ</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><msub><mi>u</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\widetilde{u}(\mathbf{s})=\sum_{j=1}^{N} \psi_{j}(\mathbf{s}) u_{j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6906em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">u</span></span><span class="svg-align" style="width:calc(100% - 0.0556em);margin-left:0.0556em;top:-3.4306em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.26em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.26em" viewBox="0 0 600 260" preserveAspectRatio="none"><path d="M200 55.538c-77 0-168 73.953-177 73.953-3 0-7
-2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0
 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0
 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128
-68.267.847-113-73.952-191-73.952z" /></svg></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.417em;vertical-align:-0.4358em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9812em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 的分布接近于具有连续定义的 SPDE 解的分布。 <code>式 (2)</code> SPDE 的解可以用方程左右两侧具有相同联合分布的每个有限维线性泛函来表征:</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mo stretchy="false">⟨</mo><mi>f</mi><mo separator="true">,</mo><mi mathvariant="script">L</mi><mi>u</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>f</mi><mo separator="true">,</mo><mi mathvariant="script">W</mi><mo stretchy="false">⟩</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(8)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\langle f, \mathcal{L} u\rangle = \langle f, \mathcal{W}\rangle \tag{8}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathcal">L</span><span class="mord mathnormal">u</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathcal" style="margin-right:0.08222em;">W</span><span class="mclose">⟩</span></span><span class="tag"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">8</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> 表示测试函数。</p>
<p>对于有限表示 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>u</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{u}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">u</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">~</span></span></span></span></span></span></span></span></span></span>  这对于泛函的任意集合都无法实现，但通过选择特定的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span> 维泛函集，可以控制近似特性。对于 <code>式(3)</code> 的 Whittle SPDE，Lindgren 等 (2011) 使用的方法是用基函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ψ</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\psi_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 作为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">α = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> 时的测试函数（一种 Galerkin 有限元方法），用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">L</mi><msub><mi>ψ</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{L} \psi_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord mathcal">L</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 作为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">α = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 时的基函数（一种最小二乘有限元方法），然后对高阶运算符应用迭代的 Galerkin 构造。与完整 SPDE 解的协方差矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">R</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal">R</span></span></span></span> 和精度矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">Q</mi></mrow><annotation encoding="application/x-tex">\mathcal{Q}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7805em;vertical-align:-0.0972em;"></span><span class="mord mathcal">Q</span></span></span></span> 乘积中的连接方式类似，这些有限元构造生成了无限维解到有限维基上的投影，使得权重向量的精度矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Q</mi></mrow><annotation encoding="application/x-tex">\mathbf{Q}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1944em;"></span><span class="mord mathbf">Q</span></span></span></span> 在模型参数中具有封闭形式的表达式。对于有限维 Hilbert 空间中权重向量为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">f</mi></mrow><annotation encoding="application/x-tex">\mathbf{f}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf" style="margin-right:0.10903em;">f</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">g</mi></mrow><annotation encoding="application/x-tex">\mathbf{g}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">g</span></span></span></span> 的函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span>，内积 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">Q</mi><mi>u</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{Q}_u( f, g)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span></span></span></span> 变为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="bold">f</mi><mi mathvariant="bold">T</mi></msup><mi mathvariant="bold">Q</mi><mi mathvariant="bold">g</mi></mrow><annotation encoding="application/x-tex">\mathbf{f^{T}Qg}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0377em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathbf" style="margin-right:0.10903em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8433em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">T</span></span></span></span></span></span></span></span></span><span class="mord mathbf" style="margin-right:0.01597em;">Qg</span></span></span></span></span>，根据构造的细节会存在微小偏差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Q</mi></mrow><annotation encoding="application/x-tex">\mathbf{Q}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1944em;"></span><span class="mord mathbf">Q</span></span></span></span>。测试函数和 SPDE 分量之间的内积可以简化为在基函数的积和基函数梯度的积上的积分，对于三角形上的分段线性基函数而言，它只涉及简单的几何。</p>
<p>令 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">C</mi></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">C</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">G</mi></mrow><annotation encoding="application/x-tex">\mathbf{G}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">G</span></span></span></span> 分别为具有元素 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">C</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mo>=</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>ψ</mi><mi>j</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\mathbf{C}_{i,j} = \langle \psi_i, \psi_j \rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9722em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">G</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mo>=</mo><mo stretchy="false">⟨</mo><mi mathvariant="normal">∇</mi><msub><mi>ψ</mi><mi>i</mi></msub><mo separator="true">,</mo><mi mathvariant="normal">∇</mi><msub><mi>ψ</mi><mi>j</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\mathbf{G}_{i, j} = \langle \nabla \psi_i, \nabla \psi_j \rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9722em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathbf">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">⟨</span><span class="mord">∇</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∇</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span></span></span></span> 的矩阵。为了说明 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">α = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">τ = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 的构造，我们为 <code>式(8)</code> 左侧得到：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mrow><mo fence="true">[</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi><mi>i</mi></msub><mo separator="true">,</mo><mi mathvariant="script">L</mi><mover accent="true"><mi>u</mi><mo>~</mo></mover><mo stretchy="false">⟩</mo><mo fence="true">]</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>N</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mrow><mo fence="true">[</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi><mi>i</mi></msub><mo separator="true">,</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo stretchy="false">(</mo><msup><mi>κ</mi><mn>2</mn></msup><mo>−</mo><mi mathvariant="normal">Δ</mi><mo stretchy="false">)</mo><msub><mi>ψ</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo><msub><mi>u</mi><mi>j</mi></msub><mo stretchy="false">⟩</mo><mo fence="true">]</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>N</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mrow><mo fence="true">[</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo stretchy="false">⟨</mo><msub><mi>ψ</mi><mi>i</mi></msub><mo separator="true">,</mo><mo stretchy="false">(</mo><msup><mi>κ</mi><mn>2</mn></msup><mo>−</mo><mi mathvariant="normal">Δ</mi><mo stretchy="false">)</mo><msub><mi>ψ</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo stretchy="false">⟩</mo><msub><mi>u</mi><mi>j</mi></msub><mo fence="true">]</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>N</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><msup><mi>κ</mi><mn>2</mn></msup><mi mathvariant="bold">C</mi><mo>+</mo><mi mathvariant="bold">G</mi><mo stretchy="false">)</mo><mi mathvariant="bold">u</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
\left [\langle\psi_i, \mathcal{L} \tilde{u} \rangle\right ]_{i=1,\ldots,N} &amp;= \left [\langle\psi_i, \sum^{N}_{j=1}(\kappa^{2} − \Delta) \psi_j(\cdot) u_j \rangle\right ]_{i=1,\ldots ,N}\\
&amp;= \left [\sum^N_{j=1} \langle\psi_i, (\kappa^{2} − \Delta) \psi_j(\cdot ) \rangle u_j \right ]_{i=1,\ldots ,N}\\ 
&amp;= (\kappa^{2} \mathbf{C} + \mathbf{G}) \mathbf{u}
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:8.98em;vertical-align:-4.24em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.74em;"><span style="top:-6.74em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathcal">L</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">u</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mclose">⟩</span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1786em;"><span style="top:-2.4003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="minner mtight">…</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.0121em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"></span></span><span style="top:-0.2484em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.24em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.74em;"><span style="top:-6.74em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">[</span></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">Δ</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">]</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.9851em;"><span style="top:-1.2365em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="minner mtight">…</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5996em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.0121em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">[</span></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">Δ</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)⟩</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">]</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.9851em;"><span style="top:-1.2365em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="minner mtight">…</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5996em;"><span></span></span></span></span></span></span></span></span><span style="top:-0.2484em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mord mathbf">C</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathbf">G</span><span class="mclose">)</span><span class="mord mathbf">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.24em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><code>式（8）</code> 右侧的协方差为：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mrow><mo fence="true">[</mo><mi mathvariant="normal">Cov</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi><mi>i</mi></msub><mo separator="true">,</mo><mi mathvariant="script">W</mi><mo stretchy="false">⟩</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi><mi>j</mi></msub><mo separator="true">,</mo><mi mathvariant="script">W</mi><mo stretchy="false">⟩</mo><mo stretchy="false">)</mo><mo fence="true">]</mo></mrow><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>N</mi></mrow></msub><mo>=</mo><mi mathvariant="bold">C</mi></mrow><annotation encoding="application/x-tex">\left [\operatorname{Cov} (\langle\psi_i, \mathcal{W} \rangle \langle \psi_j, \mathcal{W}\rangle)\right ]_{i, j=1,\ldots ,N} = \mathbf{C}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2219em;vertical-align:-0.4719em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mop"><span class="mord mathrm" style="margin-right:0.01389em;">Cov</span></span><span class="mopen">(⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathcal" style="margin-right:0.08222em;">W</span><span class="mclose">⟩</span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathcal" style="margin-right:0.08222em;">W</span><span class="mclose">⟩)</span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1425em;"><span style="top:-2.3642em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="minner mtight">…</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4719em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">C</span></span></span></span></span></p>
<p>这意味着我们需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>κ</mi><mn>2</mn></msup><mi mathvariant="bold">C</mi><mo>+</mo><mi mathvariant="bold">G</mi><mo stretchy="false">)</mo><mi mathvariant="bold">u</mi><mo>∼</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mrow><mn mathvariant="bold">0</mn><mo separator="true">,</mo><mi mathvariant="bold">C</mi></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\kappa^{2} \mathbf{C} + \mathbf{G}) \mathbf{u} \sim \mathcal{N}(\mathbf{0, C})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mord mathbf">C</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">G</span><span class="mclose">)</span><span class="mord mathbf">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.14736em;">N</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">C</span></span><span class="mclose">)</span></span></span></span>。这是当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">u</mi></mrow><annotation encoding="application/x-tex">\mathbf{u}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">u</span></span></span></span> 的精度矩阵由 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>κ</mi><mn>2</mn></msup><mi mathvariant="bold">C</mi><mo>+</mo><mi mathvariant="bold">G</mi><mo stretchy="false">)</mo><msup><mi mathvariant="bold">C</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msup><mi>κ</mi><mn>2</mn></msup><mi mathvariant="bold">C</mi><mo>+</mo><mi mathvariant="bold">G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\kappa^{2} \mathbf{C} + \mathbf{G})\mathbf{C}^{-1}(\kappa^{2} \mathbf{C} + \mathbf{G})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mord mathbf">C</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathbf">G</span><span class="mclose">)</span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mord mathbf">C</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">G</span><span class="mclose">)</span></span></span></span> 给出时得到的。</p>
<p>正如 Lindgren 等 (2011) 所讨论的那样，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">C</mi></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">C</span></span></span></span> 的逆是非稀疏的，但可以用一个包含原始矩阵行和（row-sums）的对角矩阵版本 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi mathvariant="bold">C</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{\mathbf{C}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.923em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.923em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">C</span></span><span style="top:-3.6051em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span></span></span></span> 替换 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">C</mi></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">C</span></span></span></span>，由于基函数之和为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，所以有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>C</mi><mo>~</mo></mover><mrow><mi>i</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub><mo>=</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi><mi>i</mi></msub><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\tilde{C}_{i,i} = \langle\psi_i, 1\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2063em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">⟩</span></span></span></span> 。这种 <strong>质量集中</strong> 的技术对于具有局部基函数的有限元方法很常见，但不适用于具有全局支持的基函数方法。Bakka (2019 <sup class="refplus-num"><a href="#ref-Bakka2019">[6]</a></sup>)； Lindgren 和 Rue (2008 <sup class="refplus-num"><a href="#ref-Lindgren2008">[89]</a></sup>) 提供了有关此构造的更多数学细节。</p>
<p>更一般地，为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mo>…</mo></mrow><annotation encoding="application/x-tex">α = 1, 2, 3,\ldots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">τ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span></span></span> 构造的精度矩阵由下式给出：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi mathvariant="bold">Q</mi><mo>=</mo><msup><mi>τ</mi><mn>2</mn></msup><msup><mi mathvariant="bold">C</mi><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mi>κ</mi><mn>2</mn></msup><mi mathvariant="bold">I</mi><mo>+</mo><msup><mi mathvariant="bold">C</mi><mrow><mo>−</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><mi mathvariant="bold">G</mi><msup><mi mathvariant="bold">C</mi><mrow><mo>−</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><msup><mo stretchy="false">)</mo><mi>α</mi></msup><msup><mi mathvariant="bold">C</mi><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(9)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\mathbf{Q} = \tau^2 \mathbf{C}^{1/2}(\kappa^{2} \mathbf{I} + \mathbf{C}^{-1/2} \mathbf{G} \mathbf{C}^{-1/2})^α \mathbf{C}^{1/2} \tag{9}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1944em;"></span><span class="mord mathbf">Q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1/2</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mord mathbf">I</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1/2</span></span></span></span></span></span></span></span></span><span class="mord mathbf">G</span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1/2</span></span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1/2</span></span></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>其中使用了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">C</mi></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">C</span></span></span></span> 的对角线版本。应该注意的是，这种构造适用于任何可以用三角剖分表示的紧凑流形，并且 Green 的第一个恒等式是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>f</mi><mo separator="true">,</mo><mo>−</mo><mi mathvariant="normal">Δ</mi><mi>g</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo separator="true">,</mo><mi mathvariant="normal">∇</mi><mi>g</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle f, −\Delta g \rangle = \langle \nabla f, \nabla g \rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">⟩</span></span></span></span> (在合适的边界条件下）适用于多面体流形表面，也适用于一些不可微分的函数（参见 Lindgren 等，2011 年，附录 B.3）。这种希尔伯特空间投影方法的近似特性遵循有限元方法的共同特性，并将在 <code>第 5 节</code> 中更详细地讨论。</p>
<p>如 Bolin 和 Lindgren (2013 <sup class="refplus-num"><a href="#ref-Bolin2013">[23]</a></sup>) 所示，通过对规则网格域使用高阶 B 样条基函数或小波，可以减少整体逼近误差，Liu 等 (2016 <sup class="refplus-num"><a href="#ref-Liu2016">[93]</a></sup>) 在三角剖分上实施高阶双变量样条作为基函数。实际上，在需要时增加三角形的分辨率更容易实现，并且避免了质量集中的潜在问题，对于高阶基函数应该避免这种问题。一维域是一个重要的例外，其中分段二次 B 样条基很容易实现，并且可以带来明显的改进，特别是对于每个样条节点间隔内具有不规则间隔观测的问题。分段线性基函数会导致节点和区间中点之间的边缘方差出现明显差异，类似于核卷积方法 (Simpson 等, 2012a <sup class="refplus-num"><a href="#ref-Simpson2012a">[134]</a></sup>) 所表现出的问题，高阶 B 样条平滑了条件确定性区间效应。即使对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">α = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 的情况，这也是有用的，否则人们可能期望更平滑的基函数不会增加价值。代替质量集中，应该使用完整的五对角 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">C</mi></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">C</span></span></span></span> 矩阵，并且对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">α = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> 来说，通过表示 2-谐波运算的元素 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi mathvariant="normal">Δ</mi><msub><mi>ψ</mi><mi>i</mi></msub><mo separator="true">,</mo><mi mathvariant="normal">Δ</mi><mi>p</mi><mi>s</mi><msub><mi>i</mi><mi>j</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle \Delta \psi_i, \Delta psi_j \rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">⟨</span><span class="mord">Δ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Δ</span><span class="mord mathnormal">p</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span></span></span></span>，可以用二阶矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">G</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{G}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 替换 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">G</mi><msup><mi mathvariant="bold">C</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi mathvariant="bold">G</mi></mrow><annotation encoding="application/x-tex">\mathbf{G} \mathbf{C}^{-1} \mathbf{G}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord mathbf">G</span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord mathbf">G</span></span></span></span>。</p>
<h2 id="3-实用的空间估计和推断">3 实用的空间估计和推断</h2>
<p>本节讨论为什么高斯过程的精度矩阵表示可以更好地处理观测条件，以及何时将多个组件连接到更大的统计模型中。</p>
<h3 id="3-1-含噪声观测下的条件分布">3.1 含噪声观测下的条件分布</h3>
<p>具有加性观测噪声和 Matern 协方差的简单分层高斯过程模型可以写为：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left right" columnspacing="0em 1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>y</mi><mi>i</mi></msub><mi mathvariant="normal">∣</mi><mi>u</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>∼</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><msup><mi>τ</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mi>e</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>n</mi><mi>u</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo>∼</mo><mrow><mi mathvariant="script">G</mi><mi mathvariant="script">R</mi><mi mathvariant="script">F</mi></mrow><mo stretchy="false">(</mo><msub><mi>μ</mi><mi>u</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi>ϱ</mi><mi>M</mi></msub><mo>⋅</mo><mo separator="true">,</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
y_i|u(\cdot ) &amp;\sim \mathcal{N}(u(\mathbf{s}_i), \tau^{-2} e ), i = 1,\ldots , n 
u(\cdot) &amp;\sim \mathcal{GRF}(\mu_u(\cdot), \varrho_M\cdot, \cdot ))  \\
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5241em;vertical-align:-0.5121em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0121em;"><span style="top:-3.1479em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5121em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0121em;"><span style="top:-3.1479em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathcal" style="margin-right:0.14736em;">N</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">e</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5121em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0121em;"><span style="top:-3.1479em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</span><span class="mord mathcal">R</span><span class="mord mathcal" style="margin-right:0.09931em;">F</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">ϱ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">⋅</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">⋅</span><span class="mclose">))</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5121em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>用 <code>第 2.7 节</code> 中的有限维 Hilbert 空间表示替换完整随机场给出：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi mathvariant="bold">u</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>∼</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">μ</mi><mi>u</mi></msub><mo separator="true">,</mo><msubsup><mi mathvariant="bold">Q</mi><mi>u</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>y</mi><mi>i</mi></msub><mi mathvariant="normal">∣</mi><mi mathvariant="bold">u</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>∼</mo><mi mathvariant="script">N</mi><mrow><mo fence="true">(</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>ψ</mi><mi>j</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mi>u</mi><mi>j</mi></msub><mo separator="true">,</mo><msubsup><mi>τ</mi><mi>e</mi><mrow><mo>−</mo><mn>2</mn></mrow></msubsup><mo fence="true">)</mo></mrow><mo separator="true">,</mo><mspace width="2em"></mspace><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>n</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
\mathbf{u} &amp;\sim \mathcal{N}(\boldsymbol{\mu}_u, \mathbf{Q}^{−1}_u)\\
y_i|\mathbf{u} &amp;\sim \mathcal{N} \left ( \sum^{N}_{j=1} \psi_j(\mathbf{s}_i) u_j, \tau^{-2}_e\right),\qquad  i = 1,\ldots,n  
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.0662em;vertical-align:-2.2831em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.7831em;"><span style="top:-5.7473em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord mathbf">u</span></span></span><span style="top:-3.259em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathbf">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.2831em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.7831em;"><span style="top:-5.7473em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathcal" style="margin-right:0.14736em;">N</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">μ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.259em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathcal" style="margin-right:0.14736em;">N</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">(</span></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.1132em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.2831em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>引入具有元素 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">A</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>ψ</mi><mi>j</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{A}_{i,j} = \psi_j(\mathbf{s}_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9722em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的基函数计算矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi></mrow><annotation encoding="application/x-tex">\mathbf{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">A</span></span></span></span>，完整的观测向量模型变为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="bold">y</mi><mi mathvariant="bold">∣</mi><mi mathvariant="bold">u</mi></mrow><mo>∼</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">u</mi></mrow><mo separator="true">,</mo><msubsup><mi mathvariant="bold">Q</mi><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{y|u} \sim \mathcal{N}(\mathbf{Au}, \mathbf{Q}^{-1}_e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf">y∣u</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.14736em;">N</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">Au</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">Q</mi><mi>e</mi></msub><mo>=</mo><mi mathvariant="bold">I</mi><msubsup><mi>τ</mi><mi>e</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{Q}_e = \mathbf{I} \tau^2_e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0611em;vertical-align:-0.247em;"></span><span class="mord mathbf">I</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.1132em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> 是观测噪声的精度矩阵。通过在多元分布中使用精度矩阵版本的条件（Rue 和 Held，2005 <sup class="refplus-num"><a href="#ref-Rue2005">[119]</a></sup>），在给定观测的情况下，场的基函数权重的条件分布为：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold">u</mi><mo>∣</mo><mi mathvariant="bold">y</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>∼</mo><mi mathvariant="normal">N</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold-italic">μ</mi><mrow><mi>u</mi><mo>∣</mo><mi>y</mi></mrow></msub><mo separator="true">,</mo><msubsup><mi mathvariant="bold">Q</mi><mrow><mi>u</mi><mo>∣</mo><mi>y</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo fence="true">)</mo></mrow><mo separator="true">,</mo></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi mathvariant="bold">Q</mi><mrow><mi>u</mi><mo>∣</mo><mi>y</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold">Q</mi><mi>u</mi></msub><mo>+</mo><msup><mi mathvariant="bold">A</mi><mi mathvariant="normal">⊤</mi></msup><msub><mi mathvariant="bold">Q</mi><mi>e</mi></msub><mi mathvariant="bold">A</mi><mo separator="true">,</mo></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>μ</mi><mrow><mi>u</mi><mo>∣</mo><mi>y</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi mathvariant="bold-italic">μ</mi><mi>u</mi></msub><mo>+</mo><msubsup><mi mathvariant="bold">Q</mi><mrow><mi>u</mi><mo>∣</mo><mi>y</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><msup><mi mathvariant="bold">A</mi><mi mathvariant="normal">⊤</mi></msup><msub><mi mathvariant="bold">Q</mi><mi>e</mi></msub><mrow><mo fence="true">(</mo><mi mathvariant="bold">y</mi><mo>−</mo><mi mathvariant="bold">A</mi><msub><mi mathvariant="bold-italic">μ</mi><mi>u</mi></msub><mo fence="true">)</mo></mrow></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
\mathbf{u} \mid \mathbf{y} &amp; \sim \mathrm{N}\left(\boldsymbol{\mu}_{u \mid y}, \mathbf{Q}_{u \mid y}^{-1}\right), \tag{10}\\
\mathbf{Q}_{u \mid y} &amp; =\mathbf{Q}_u+\mathbf{A}^{\top} \mathbf{Q}_e \mathbf{A}, \tag{11}\\
\mu_{u \mid y} &amp; =\boldsymbol{\mu}_u+\mathbf{Q}_{u \mid y}^{-1} \mathbf{A}^{\top} \mathbf{Q}_e\left(\mathbf{y}-\mathbf{A} \boldsymbol{\mu}_u\right) \tag{12} 
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.3636em;vertical-align:-2.4318em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9318em;"><span style="top:-4.9318em;"><span class="pstrut" style="height:3.15em;"></span><span class="mord"><span class="mord mathbf">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span></span><span style="top:-3.0827em;"><span class="pstrut" style="height:3.15em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mrel mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.5235em;"><span class="pstrut" style="height:3.15em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mrel mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4318em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9318em;"><span style="top:-4.9318em;"><span class="pstrut" style="height:3.15em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathrm">N</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">μ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2809em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mrel mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4191em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.3697em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mrel mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5053em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span></span></span><span style="top:-3.0827em;"><span class="pstrut" style="height:3.15em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⊤</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathbf">A</span><span class="mpunct">,</span></span></span><span style="top:-1.5235em;"><span class="pstrut" style="height:3.15em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">μ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.3697em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mrel mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5053em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⊤</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathbf">A</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">μ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4318em;"><span></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9318em;"><span style="top:-4.9318em;"><span class="pstrut" style="height:3.15em;"></span><span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">10</span></span><span class="mord">)</span></span></span></span><span style="top:-3.0827em;"><span class="pstrut" style="height:3.15em;"></span><span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">11</span></span><span class="mord">)</span></span></span></span><span style="top:-1.5235em;"><span class="pstrut" style="height:3.15em;"></span><span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">12</span></span><span class="mord">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4318em;"><span></span></span></span></span></span></span></span></span></p>
<p>这些方程提供了该场的克里金估计的有限希尔伯特空间表示，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><msub><mi>ψ</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><msub><mrow><mo fence="true">[</mo><msub><mi mathvariant="bold-italic">μ</mi><mrow><mi>u</mi><mi mathvariant="normal">∣</mi><mi>y</mi></mrow></msub><mo fence="true">]</mo></mrow><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\sum^N_{j=1} \psi_j(\mathbf{s}) \left [\boldsymbol{\mu}_{u|y} \right ]_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.9858em;vertical-align:-0.8358em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9812em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">[</span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">μ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2809em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mord mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4191em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">]</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.2381em;"><span style="top:-2.0003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8358em;"><span></span></span></span></span></span></span></span></span></span>。对于局部支持的基函数，条件精度矩阵仍然是稀疏的，计算成本较低，并且条件期望仅涉及对该稀疏矩阵的线性求解。通过由矩阵稀疏模式生成的马尔可夫图的自动重排序，直接的 Cholesky 分解可以保留高度的稀疏性，使其成为理想的直接求解方法。此外，通过对后验精度矩阵的 Cholesky 分解实施 Takahashi 递归（Takahashi 等，1973 年，Erisman 和 Tinney，1975 年、Rue 和 Martino，2007 年、Rue 和 Held，2010 年），可以快速得出后验边缘方差和邻域协方差，例如 <code>R-INLA</code> 包中的 <code>inla.qinv()</code> 函数实现，整个过程无需计算密集矩阵的逆。</p>
<p>如果我们在这个模型中有未知的（超）参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">θ</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\theta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span></span></span></span>，例如边缘方差或变程，那么我们可以直接计算后验密度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold">y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi (\boldsymbol{\theta}|\mathbf{y})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="mord">∣</span><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="mclose">)</span></span></span></span>，比如:</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>π</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold">y</mi><mo stretchy="false">)</mo><mo>∝</mo><mfrac><mrow><mi>π</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><mi>π</mi><mo stretchy="false">(</mo><mi mathvariant="bold">u</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><mi>π</mi><mo stretchy="false">(</mo><mi mathvariant="bold">y</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold">u</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo></mrow><mrow><mi>π</mi><mo stretchy="false">(</mo><mi mathvariant="bold">u</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold">y</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\pi (\boldsymbol{\theta} | \mathbf{y})  \propto  \frac{\pi(\boldsymbol{\theta}) \pi(\mathbf{u} | \boldsymbol{\theta}) \pi (\mathbf{y} | \mathbf{u}, \boldsymbol{\theta})}{\pi (\mathbf{u} | \mathbf{y}, \boldsymbol{\theta})}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="mord">∣</span><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mopen">(</span><span class="mord mathbf">u</span><span class="mord">∣</span><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mopen">(</span><span class="mord mathbf">u</span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="mord">∣</span><span class="mord mathbf">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi><mo stretchy="false">(</mo><mi mathvariant="bold">u</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold">y</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi(\mathbf{u}| \mathbf{y}, \boldsymbol{\theta})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mopen">(</span><span class="mord mathbf">u</span><span class="mord">∣</span><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="mclose">)</span></span></span></span> 是高斯分布的。除了已计算的条件均值和精度矩阵之外，唯一新进入的项是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo><mi mathvariant="normal">∣</mi><msub><mi mathvariant="bold">Q</mi><mrow><mi>u</mi><mi mathvariant="normal">∣</mi><mi>y</mi></mrow></msub><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">\log |\mathbf{Q}_{u|y}|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1052em;vertical-align:-0.3552em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mord mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span></span>，可直接从其 Cholesky 分解中获得。</p>
<h3 id="3-2-添加模型组件">3.2 添加模型组件</h3>
<p>我们的目标不仅是处理<code>第 3.1 节</code>中讨论的简单模型构造，而且还希望处高斯模型组份之和形成的线性预测 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">η</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{η}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span></span></span></span>（Rue 等，2009 <sup class="refplus-num"><a href="#ref-Rue2009">[122]</a></sup>）。例如，我们考虑 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">η</mi><mo>=</mo><msub><mi mathvariant="bold">A</mi><mi>u</mi></msub><mi mathvariant="bold">u</mi><mo>+</mo><msub><mi mathvariant="bold">A</mi><mi>v</mi></msub><mi mathvariant="bold">v</mi><mo>+</mo><msub><mi mathvariant="bold">A</mi><mi>w</mi></msub><mi mathvariant="bold">w</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{η} = \mathbf{A}_u \mathbf{u} + \mathbf{A}_v\mathbf{v} + \mathbf{A}_w\mathbf{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathbf">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathbf" style="margin-right:0.01597em;">w</span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">u</mi><mo>∼</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mn mathvariant="bold">0</mn><mo separator="true">,</mo><msubsup><mi mathvariant="bold">Q</mi><mi>u</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{u} \sim \mathcal{N}(\mathbf{0}, \mathbf{Q}^{-1}_u)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.14736em;">N</span><span class="mopen">(</span><span class="mord mathbf">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">v</mi><mo>∼</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mn mathvariant="bold">0</mn><mo separator="true">,</mo><msubsup><mi mathvariant="bold">Q</mi><mi>v</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{v} \sim \mathcal{N}(\mathbf{0}, \mathbf{Q}^{-1}_v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.14736em;">N</span><span class="mopen">(</span><span class="mord mathbf">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>,  <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi><mo>∼</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mn mathvariant="bold">0</mn><mo separator="true">,</mo><msubsup><mi mathvariant="bold">Q</mi><mi>w</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">w \sim \mathcal{N}(\mathbf{0}, \mathbf{Q}^{-1}_w)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.14736em;">N</span><span class="mopen">(</span><span class="mord mathbf">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">A</mi><mi>u</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{A}_u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">A</mi><mi>v</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{A}_v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">A</mi><mi>w</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{A}_w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 分别是将 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">u</mi></mrow><annotation encoding="application/x-tex">\mathbf{u}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">u</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">v</mi></mrow><annotation encoding="application/x-tex">\mathbf{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">w</mi></mrow><annotation encoding="application/x-tex">\mathbf{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">w</span></span></span></span> 中的（隐）变量连接到线性预测 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">η</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{η}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span></span></span></span> 的矩阵。隐模型组件可能包含 “有限维 SPDE 表示”、“固定效应系数”和其他结构化或非结构化随机效应。通过精度矩阵添加模型组件的一个重要性质是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">η</mi><mo separator="true">,</mo><mrow><mi mathvariant="bold">u</mi><mo separator="true">,</mo><mi mathvariant="bold">v</mi><mo separator="true">,</mo><mi mathvariant="bold">w</mi></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\boldsymbol{η},\mathbf{ u, v, w})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">w</span></span><span class="mclose">)</span></span></span></span> 的联合精度矩阵的结构直接可用。这一点特别重要，因为如果有协方差参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">θ</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\theta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span></span></span></span>，当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">θ</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\theta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span></span></span></span> 的某些元素发生变化时，我们不需要重建整个联合精度矩阵，而只需重新计算直接受影响的元素即可。</p>
<p>我们可以通过多种方式进行计算，下面重点讨论其中三种：</p>
<p><strong>（1）策略一</strong></p>
<p>第一种策略是直接使用模型组件的联合精度矩阵，并在根据观测调节模型时，确定性地形成线性预测 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">η</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{η}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span></span></span></span>。此时，联合精度矩阵和预测可以写成：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Prec</mi><mo>⁡</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">u</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">v</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">w</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="left left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi mathvariant="bold">Q</mi><mi>u</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi mathvariant="bold">Q</mi><mi>v</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi mathvariant="bold">Q</mi><mi>w</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo separator="true">,</mo><mspace width="1em"></mspace><mtext>&nbsp;and&nbsp;</mtext><mspace width="1em"></mspace><mi mathvariant="bold-italic">η</mi><mo>=</mo><mi mathvariant="bold">A</mi><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">u</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">v</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">w</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\operatorname{Prec}\left[\begin{array}{c}
\mathbf{u} \\
\mathbf{v} \\
\mathbf{w}
\end{array}\right]=\left[\begin{array}{lll}
\mathbf{Q}_u &amp; &amp; \\
&amp; \mathbf{Q}_v &amp; \\
&amp; &amp; \mathbf{Q}_w
\end{array}\right], \quad \text { and } \quad \boldsymbol{\eta}=\mathbf{A}\left[\begin{array}{c}
\mathbf{u} \\
\mathbf{v} \\
\mathbf{w}
\end{array}\right] 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="mop"><span class="mord mathrm">Prec</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">u</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">v</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">&nbsp;and&nbsp;</span></span><span class="mspace" style="margin-right:1em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="mord mathbf">A</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">u</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">v</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>使用组合矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="left left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi mathvariant="bold">A</mi><mi>u</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi mathvariant="bold">A</mi><mi>v</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi mathvariant="bold">A</mi><mi>w</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\mathbf{A}=\left[\begin{array}{lll}\mathbf{A}_u &amp; \mathbf{A}_v &amp; \mathbf{A}_w\end{array}\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span></span>。有了这个公式，我们然后将<code>式（11）</code>和<code>式（12）</code>直接应用于联合组件模型。</p>
<p><strong>（2）策略二</strong></p>
<p>第二种策略是通过添加具有高精度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">τ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span></span></span> 的小噪声项，为线性预测建立近似联合精度。通过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">η</mi><mo>=</mo><mi mathvariant="bold">u</mi><mo>+</mo><mi mathvariant="bold">v</mi><mo>+</mo><mi mathvariant="bold">w</mi><mo>+</mo><msup><mi>τ</mi><mrow><mo>−</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><mi mathvariant="bold-italic">ϵ</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\eta}=\mathbf{u}+\mathbf{v}+\mathbf{w}+\tau^{-1 / 2} \boldsymbol{\epsilon}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathbf">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">w</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1/2</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol">ϵ</span></span></span></span></span></span> 其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">ϵ</mi><mo>∼</mo><mi mathvariant="normal">N</mi><mo stretchy="false">(</mo><mn mathvariant="bold">0</mn><mo separator="true">,</mo><mi mathvariant="bold">I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\boldsymbol{\epsilon} \sim \mathrm{N}(\mathbf{0}, \mathbf{I})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">ϵ</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">N</span><span class="mopen">(</span><span class="mord mathbf">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">I</span><span class="mclose">)</span></span></span></span>，我们得到</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Prec</mi><mo>⁡</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold-italic">η</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">u</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">v</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">w</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="left left left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn mathvariant="bold">0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi mathvariant="bold">Q</mi><mi>u</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi mathvariant="bold">Q</mi><mi>v</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi mathvariant="bold">Q</mi><mi>w</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>+</mo><mi>τ</mi><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">I</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msubsup><mi mathvariant="bold">A</mi><mi>u</mi><mi mathvariant="normal">⊤</mi></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msubsup><mi mathvariant="bold">A</mi><mi>v</mi><mi mathvariant="normal">⊤</mi></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msubsup><mi mathvariant="bold">A</mi><mi>w</mi><mi mathvariant="normal">⊤</mi></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="left left left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">I</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi mathvariant="bold">A</mi><mi>u</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi mathvariant="bold">A</mi><mi>v</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi mathvariant="bold">A</mi><mi>w</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\operatorname{Prec}\left[\begin{array}{c}
\boldsymbol{\eta} \\
\mathbf{u} \\
\mathbf{v} \\
\mathbf{w}
\end{array}\right]=\left[\begin{array}{llll}
\mathbf{0} &amp; &amp; &amp; \\
&amp; \mathbf{Q}_u &amp; &amp; \\
&amp; &amp; \mathbf{Q}_v &amp; \\
&amp; &amp; &amp; \mathbf{Q}_w
\end{array}\right]+\tau\left[\begin{array}{c}
\mathbf{I} \\
-\mathbf{A}_u^{\top} \\
-\mathbf{A}_v^{\top} \\
-\mathbf{A}_w^{\top}
\end{array}\right]\left[\begin{array}{llll}
\mathbf{I} &amp; -\mathbf{A}_u &amp; -\mathbf{A}_v &amp; -\mathbf{A}_w
\end{array}\right] 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.8em;vertical-align:-2.15em;"></span><span class="mop"><span class="mord mathrm">Prec</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.65em;"><span class="pstrut" style="height:6.8em;"></span><span style="width:0.667em;height:4.800em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="4.800em" viewBox="0 0 667 4800"><path d="M403 1759 V84 H666 V0 H319 V1759 v1200 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v1200 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">u</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">v</span></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.65em;"><span class="pstrut" style="height:6.8em;"></span><span style="width:0.667em;height:4.800em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="4.800em" viewBox="0 0 667 4800"><path d="M347 1759 V0 H0 V84 H263 V1759 v1200 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v1200 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:4.8em;vertical-align:-2.15em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.65em;"><span class="pstrut" style="height:6.8em;"></span><span style="width:0.667em;height:4.800em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="4.800em" viewBox="0 0 667 4800"><path d="M403 1759 V84 H666 V0 H319 V1759 v1200 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v1200 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">0</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.65em;"><span class="pstrut" style="height:6.8em;"></span><span style="width:0.667em;height:4.800em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="4.800em" viewBox="0 0 667 4800"><path d="M347 1759 V0 H0 V84 H263 V1759 v1200 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v1200 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:4.8273em;vertical-align:-2.1637em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.65em;"><span class="pstrut" style="height:6.8em;"></span><span style="width:0.667em;height:4.800em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="4.800em" viewBox="0 0 667 4800"><path d="M403 1759 V84 H666 V0 H319 V1759 v1200 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v1200 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6637em;"><span style="top:-4.8237em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">I</span></span></span><span style="top:-3.6146em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⊤</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.4054em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⊤</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.1963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⊤</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1637em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.65em;"><span class="pstrut" style="height:6.8em;"></span><span style="width:0.667em;height:4.800em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="4.800em" viewBox="0 0 667 4800"><path d="M347 1759 V0 H0 V84 H263 V1759 v1200 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v1200 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span></span></span></p>
<p>请注意，我们需要保留所有模型组件，因为边缘化会破坏马尔可夫特性。</p>
<p><strong>（3）策略三</strong></p>
<p>第三种策略是使用累积和。这种方法可以应用于 “组合效应能够用通用表示形式编写的” 模型，例如 SPDE 模型在不同空间分辨率下的相等或嵌套三角剖分。假设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">B</mi><mrow><mi>u</mi><mi>v</mi></mrow></msub><mi mathvariant="bold">v</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}_{u v}\mathbf{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">uv</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span></span></span></span> 从粗略的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span> 基函数转换为更精细的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> 基函数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">B</mi><mrow><mi>v</mi><mi>w</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}_{v w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 也类似。这使得线性预测可以表示为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">η</mi><mo>=</mo><msub><mi mathvariant="bold">A</mi><mi>u</mi></msub><mrow><mo fence="true">{</mo><mi mathvariant="bold">u</mi><mo>+</mo><msub><mi mathvariant="bold">B</mi><mrow><mi>u</mi><mi>v</mi></mrow></msub><mrow><mo fence="true">(</mo><mi mathvariant="bold">v</mi><mo>+</mo><msub><mi mathvariant="bold">B</mi><mrow><mi>v</mi><mi>w</mi></mrow></msub><mi mathvariant="bold">w</mi><mo fence="true">)</mo></mrow><mo fence="true">}</mo></mrow></mrow><annotation encoding="application/x-tex">\boldsymbol{\eta}=\mathbf{A}_u\left\{\mathbf{u}+\mathbf{B}_{u v}\left(\mathbf{v} +\mathbf{B}_{v w} \mathbf{w}\right)\right\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">{</span><span class="mord mathbf">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">uv</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathbf" style="margin-right:0.01597em;">v</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathbf" style="margin-right:0.01597em;">w</span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mclose delimcenter" style="top:0em;">}</span></span></span></span></span>。然后我们可以定义 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi mathvariant="bold">v</mi><mo stretchy="true">~</mo></mover><mo>=</mo><mi mathvariant="bold">v</mi><mo>+</mo><msub><mi mathvariant="bold">B</mi><mrow><mi>v</mi><mi>w</mi></mrow></msub><mi mathvariant="bold">w</mi></mrow><annotation encoding="application/x-tex">\widetilde{\mathbf{v}}=\mathbf{v}+\mathbf{B}_{v w} \mathbf{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7044em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7044em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span></span><span class="svg-align" style="width:calc(100% - 0.0556em);margin-left:0.0556em;top:-3.4444em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.26em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.26em" viewBox="0 0 600 260" preserveAspectRatio="none"><path d="M200 55.538c-77 0-168 73.953-177 73.953-3 0-7
-2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0
 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0
 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128
-68.267.847-113-73.952-191-73.952z" /></svg></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathbf" style="margin-right:0.01597em;">w</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi mathvariant="bold">u</mi><mo stretchy="true">~</mo></mover><mo>=</mo><mi mathvariant="bold">u</mi><mo>+</mo><msub><mi mathvariant="bold">B</mi><mrow><mi>u</mi><mi>v</mi></mrow></msub><mover accent="true"><mi mathvariant="bold">v</mi><mo stretchy="true">~</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde{\mathbf{u}}= \mathbf {u}+\mathbf{B}_{u v} \widetilde{\mathbf{v}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7044em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7044em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">u</span></span><span class="svg-align" style="width:calc(100% - 0.0556em);margin-left:0.0556em;top:-3.4444em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.26em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.26em" viewBox="0 0 600 260" preserveAspectRatio="none"><path d="M200 55.538c-77 0-168 73.953-177 73.953-3 0-7
-2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0
 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0
 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128
-68.267.847-113-73.952-191-73.952z" /></svg></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathbf">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8544em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">uv</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7044em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span></span><span class="svg-align" style="width:calc(100% - 0.0556em);margin-left:0.0556em;top:-3.4444em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.26em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.26em" viewBox="0 0 600 260" preserveAspectRatio="none"><path d="M200 55.538c-77 0-168 73.953-177 73.953-3 0-7
-2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0
 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0
 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128
-68.267.847-113-73.952-191-73.952z" /></svg></span></span></span></span></span></span></span></span></span>，使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">w</mi><mo>∼</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mn mathvariant="bold">0</mn><mo separator="true">,</mo><msubsup><mi mathvariant="bold">Q</mi><mi>w</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{w} \sim \mathcal{N}(\mathbf{0} , \mathbf {Q}_w^{-1})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">w</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.14736em;">N</span><span class="mopen">(</span><span class="mord mathbf">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi mathvariant="bold">v</mi><mo stretchy="true">~</mo></mover><mo>∣</mo><mi mathvariant="bold">w</mi><mo>∼</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">B</mi><mrow><mi>v</mi><mi>w</mi></mrow></msub><mi mathvariant="bold">w</mi><mo separator="true">,</mo><msubsup><mi mathvariant="bold">Q</mi><mi>v</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widetilde{\mathbf{v}} \mid \mathbf{w} \sim \mathcal{N}(\mathbf{B}_{ v w} \mathbf {w}, \mathbf{Q}_v^{-1})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7044em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span></span><span class="svg-align" style="width:calc(100% - 0.0556em);margin-left:0.0556em;top:-3.4444em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.26em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.26em" viewBox="0 0 600 260" preserveAspectRatio="none"><path d="M200 55.538c-77 0-168 73.953-177 73.953-3 0-7
-2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0
 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0
 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128
-68.267.847-113-73.952-191-73.952z" /></svg></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">w</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.14736em;">N</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathbf" style="margin-right:0.01597em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi mathvariant="bold">u</mi><mo stretchy="true">~</mo></mover><mo>∣</mo><mover accent="true"><mi mathvariant="bold">v</mi><mo stretchy="true">~</mo></mover><mo separator="true">,</mo><mi mathvariant="bold">w</mi><mo>∼</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">B</mi><mrow><mi>u</mi><mi>v</mi></mrow></msub><mover accent="true"><mi mathvariant="bold">v</mi><mo stretchy="true">~</mo></mover><mo separator="true">,</mo><msubsup><mi mathvariant="bold">Q</mi><mi>u</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widetilde{\mathbf{u}} \mid \widetilde{\mathbf{v}}, \mathbf{w} \sim \mathcal{ N}(\mathbf{B}_{u v} \widetilde{\mathbf{v}}, \mathbf{Q}_u^{-1})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7044em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">u</span></span><span class="svg-align" style="width:calc(100% - 0.0556em);margin-left:0.0556em;top:-3.4444em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.26em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.26em" viewBox="0 0 600 260" preserveAspectRatio="none"><path d="M200 55.538c-77 0-168 73.953-177 73.953-3 0-7
-2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0
 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0
 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128
-68.267.847-113-73.952-191-73.952z" /></svg></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8989em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7044em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span></span><span class="svg-align" style="width:calc(100% - 0.0556em);margin-left:0.0556em;top:-3.4444em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.26em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.26em" viewBox="0 0 600 260" preserveAspectRatio="none"><path d="M200 55.538c-77 0-168 73.953-177 73.953-3 0-7
-2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0
 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0
 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128
-68.267.847-113-73.952-191-73.952z" /></svg></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">w</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.14736em;">N</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">uv</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7044em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span></span><span class="svg-align" style="width:calc(100% - 0.0556em);margin-left:0.0556em;top:-3.4444em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.26em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.26em" viewBox="0 0 600 260" preserveAspectRatio="none"><path d="M200 55.538c-77 0-168 73.953-177 73.953-3 0-7
-2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0
 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0
 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128
-68.267.847-113-73.952-191-73.952z" /></svg></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。则联合精度矩阵为：</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">Prec</mi><mo>⁡</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mover accent="true"><mi mathvariant="bold">u</mi><mo>~</mo></mover></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mover accent="true"><mi mathvariant="bold">v</mi><mo stretchy="true">~</mo></mover></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">w</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn mathvariant="bold">0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn mathvariant="bold">0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">I</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><msub><mi mathvariant="bold">Q</mi><mi>w</mi></msub><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="left left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn mathvariant="bold">0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn mathvariant="bold">0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">I</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>+</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn mathvariant="bold">0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">I</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msubsup><mi mathvariant="bold">B</mi><mrow><mi>v</mi><mi>w</mi></mrow><mi mathvariant="normal">⊤</mi></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><msub><mi mathvariant="bold">Q</mi><mi>v</mi></msub><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="left left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn mathvariant="bold">0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">I</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi mathvariant="bold">B</mi><mrow><mi>v</mi><mi>w</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>+</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">I</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msubsup><mi mathvariant="bold">B</mi><mrow><mi>u</mi><mi>v</mi></mrow><mi mathvariant="normal">⊤</mi></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn mathvariant="bold">0</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><msub><mi mathvariant="bold">Q</mi><mi>u</mi></msub><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="left left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="bold">I</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi mathvariant="bold">B</mi><mrow><mi>u</mi><mi>v</mi></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn mathvariant="bold">0</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi mathvariant="bold">Q</mi><mi>u</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi mathvariant="bold">Q</mi><mi>u</mi></msub><msub><mi mathvariant="bold">B</mi><mrow><mi>u</mi><mi>v</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msubsup><mi mathvariant="bold">B</mi><mrow><mi>u</mi><mi>v</mi></mrow><mi mathvariant="normal">⊤</mi></msubsup><msub><mi mathvariant="bold">Q</mi><mi>u</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi mathvariant="bold">Q</mi><mi>v</mi></msub><mo>+</mo><msubsup><mi mathvariant="bold">B</mi><mi>u</mi><mi mathvariant="normal">⊤</mi></msubsup><msub><mi mathvariant="bold">Q</mi><mi>u</mi></msub><msub><mi mathvariant="bold">B</mi><mrow><mi>u</mi><mi>v</mi></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi mathvariant="bold">Q</mi><mi>v</mi></msub><msub><mi mathvariant="bold">B</mi><mrow><mi>v</mi><mi>w</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msubsup><mi mathvariant="bold">B</mi><mrow><mi>v</mi><mi>w</mi></mrow><mi>ν</mi></msubsup><msub><mi mathvariant="bold">Q</mi><mi>v</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi mathvariant="bold">Q</mi><mi>w</mi></msub><mo>+</mo><msubsup><mi mathvariant="bold">B</mi><mrow><mi>v</mi><mi>w</mi></mrow><mi mathvariant="normal">⊤</mi></msubsup><msub><mi mathvariant="bold">Q</mi><mi>v</mi></msub><msub><mi mathvariant="bold">B</mi><mrow><mi>v</mi><mi>w</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mi mathvariant="normal">.</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\operatorname{Prec}\left[\begin{array}{c}
\tilde{\mathbf{u}} \\
\widetilde{\mathbf{v}} \\
\mathbf{w}
\end{array}\right] &amp; =\left[\begin{array}{l}
\mathbf{0} \\
\mathbf{0} \\
\mathbf{I}
\end{array}\right] \mathbf{Q}_w\left[\begin{array}{lll}
\mathbf{0} &amp; \mathbf{0} &amp; \mathbf{I}
\end{array}\right]+\left[\begin{array}{c}
\mathbf{0} \\
\mathbf{I} \\
-\mathbf{B}_{v w}^{\top}
\end{array}\right] \mathbf{Q}_v\left[\begin{array}{lll}
\mathbf{0} &amp; \mathbf{I} &amp; -\mathbf{B}_{v w}
\end{array}\right]+\left[\begin{array}{cc}
\mathbf{I} \\
-\mathbf{B}_{u v}^{\top} \\
\mathbf{0}
\end{array}\right] \mathbf{Q}_u\left[\begin{array}{lll}
\mathbf{I} &amp; -\mathbf{B}_{u v} &amp; \mathbf{0}
\end{array}\right] \\
&amp; =\left[\begin{array}{ccc}
\mathbf{Q}_u &amp; -\mathbf{Q}_u \mathbf{B}_{u v} \\
-\mathbf{B}_{u v}^{\top} \mathbf{Q}_u &amp; \mathbf{Q}_v+\mathbf{B}_u^{\top} \mathbf{Q}_u \mathbf{B}_{u v} &amp; -\mathbf{Q}_v \mathbf{B}_{v w} \\
&amp; -\mathbf{B}_{v w}^\nu \mathbf{Q}_v &amp; \mathbf{Q}_w+\mathbf{B}_{v w}^{\top} \mathbf{Q}_v \mathbf{B}_{v w}
\end{array}\right] .
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.8273em;vertical-align:-3.6637em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1637em;"><span style="top:-6.1682em;"><span class="pstrut" style="height:4.0591em;"></span><span class="mord"><span class="mop"><span class="mord mathrm">Prec</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">u</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">~</span></span></span></span></span></span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7044em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span></span><span class="svg-align" style="width:calc(100% - 0.0556em);margin-left:0.0556em;top:-3.4444em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.26em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.26em" viewBox="0 0 600 260" preserveAspectRatio="none"><path d="M200 55.538c-77 0-168 73.953-177 73.953-3 0-7
-2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0
 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0
 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128
-68.267.847-113-73.952-191-73.952z" /></svg></span></span></span></span></span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-2.2546em;"><span class="pstrut" style="height:4.0591em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6637em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1637em;"><span style="top:-6.1682em;"><span class="pstrut" style="height:4.0591em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">0</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">0</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0546em;"><span style="top:-4.2146em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">0</span></span></span><span style="top:-3.0146em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">I</span></span></span><span style="top:-1.8054em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⊤</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5546em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0546em;"><span style="top:-4.2146em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">I</span></span></span><span style="top:-3.0054em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">uv</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⊤</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.8054em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5546em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">uv</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span><span style="top:-2.2546em;"><span class="pstrut" style="height:4.0591em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0591em;"><span style="top:-4.2191em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">uv</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⊤</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.8009em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5591em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0591em;"><span style="top:-4.2191em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">uv</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⊤</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">uv</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.8009em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5591em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8591em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.8009em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⊤</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5591em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v0 v1759 h84z" /></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">.</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6637em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>此构造可以与第一种或第二种策略结合使用，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">η</mi><mo>=</mo><msub><mi mathvariant="bold">A</mi><mi>u</mi></msub><mover accent="true"><mi mathvariant="bold">u</mi><mo>~</mo></mover><mo>+</mo><mn mathvariant="bold">0</mn><mover accent="true"><mi mathvariant="bold">v</mi><mo>~</mo></mover><mo>+</mo><mn mathvariant="bold">0</mn><mi mathvariant="bold">w</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{η} = \mathbf{A}_u \tilde{\mathbf{u}} + \mathbf{0} \tilde{\mathbf{v}} + \mathbf{0} \mathbf{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">u</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7646em;vertical-align:-0.0833em;"></span><span class="mord mathbf">0</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">0w</span></span></span></span>，因此线性预测器仅直接连接到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi mathvariant="bold">u</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{\mathbf{u}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6813em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">u</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">~</span></span></span></span></span></span></span></span></span></span>。累积方法非常优雅和高效，也可以作为迭代求解器的多尺度预处理器的基础。一个潜在的缺点是一些原始模型组件没有明确出现。但当目的是推断 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">θ</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\theta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span></span></span></span> 或模型组件并非直接感兴趣时，这不是问题。</p>
<p>如上所示，联合精度矩阵是直接可用的，不需要在协方差矩阵参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">θ</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\theta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span></span></span></span> 的值发生变化时重建。 <code>R-INLA</code> (<a target="_blank" rel="noopener" href="http://www.r-inla.org">http://www.r-inla.org</a>) 实现目前将所有这三种策略混合用于各种模型组件和联合模型。</p>
<h3 id="3-3-贝叶斯推断和非高斯观测">3.3 贝叶斯推断和非高斯观测</h3>
<p>当 SPDE 模型（或多个模型）用于更大的 GLM 模型时，例如泊松计数数据，</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>y</mi><mi>i</mi></msub><mi mathvariant="normal">∣</mi><msub><mi>η</mi><mi>i</mi></msub><mo>∼</mo><mrow><mi mathvariant="script">P</mi><mi mathvariant="script">o</mi><mi mathvariant="script">i</mi><mi mathvariant="script">s</mi><mi mathvariant="script">s</mi><mi mathvariant="script">o</mi><mi mathvariant="script">n</mi></mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mi>i</mi></msub><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>η</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y_i | η_i \sim \mathcal{Poisson}(E_i \exp(η_i))
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span><span class="mord mathnormal">o</span><span class="mord mathnormal">i</span><span class="mord mathnormal">sso</span><span class="mord mathnormal">n</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>E</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">E_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 为正常数。条件分布不再以封闭形式提供，就像其在高斯分布观测的情况下一样。确定性推断计算仍然是可能的，但需要一些近似。有了表现良好的类高斯结构模型，我们可以使用集成嵌套拉普拉斯近似 (INLA)，使近似的影响远小于估计本身的不确定性。简而言之，INLA 要求以嵌套方式重复 <code>第 3.1 节</code> 中概述的计算，以提供所有模型参数的后验边缘近似，不过，会使用对数似然的二阶泰勒近似，而不是 <code>式（11）</code> 的高斯情况下的项 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="bold">A</mi><mi>T</mi></msup><msub><mi mathvariant="bold">Q</mi><mi>e</mi></msub><mi mathvariant="bold">A</mi></mrow><annotation encoding="application/x-tex">\mathbf{A}^T \mathbf{Q}_e \mathbf{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0358em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathbf">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathbf">A</span></span></span></span>。因此，计算效率至关重要。Rue 等 (2009 <sup class="refplus-num"><a href="#ref-Rue2009">[122]</a></sup>) 介绍了 INLA 方法，Martins 等 (2013 <sup class="refplus-num"><a href="#ref-Martins2013">[96]</a></sup>) 讨论一些改进，Rue 等 (2017 <sup class="refplus-num"><a href="#ref-Rue2017">[123]</a></sup>) 回顾该方法，但重点关注其中的基本思想，van Niekerk 等 (2021 <sup class="refplus-num"><a href="#ref-vanNiekerk2021">[148]</a></sup>) 描述了 R-INLA 包中的一些最新扩展，而 Krainski 等(2018 <sup class="refplus-num"><a href="#ref-Krainski2018">[80]</a></sup>) 的书提供了使用 SPDE 模型和 R-INLA 包的实用指南（带代码）。</p>
<p>SPDE 模型中 (对数)变程和 (对数)边缘方差的先验设置非常重要，因为这些参数不能在填充渐近下一致地被估计（Zhang，2004 <sup class="refplus-num"><a href="#ref-Zhang2004">[166]</a></sup>）。Fuglstad 等 (2019 <sup class="refplus-num"><a href="#ref-Fuglstad2019">[60]</a></sup>) 为其推导出了带复杂性惩罚的联合先验 (Simpson 等, 2017 <sup class="refplus-num"><a href="#ref-Simpson2017">[136]</a></sup>)，并讨论了如何处理非平稳模型。此先验族是我们推荐的，在实践中效果很好。</p>
<h2 id="4-重要扩展">4 重要扩展</h2>
<h3 id="4-1-一般分数幂的方法">4.1 一般分数幂的方法</h3>
<h3 id="4-2-非高斯模型">4.2 非高斯模型</h3>
<h3 id="4-3-时空过程">4.3 时空过程</h3>
<h2 id="5-理论保证">5 理论保证</h2>
<p>在介绍了 SPDE 方法及其扩展的主要思想之后，我们现在准备研究基于 SPDE 的模型和相关计算方法的一些更具技术性的理论特性。人们可以很容易地争辩说，无论是从理论还是实践的角度来看，基于 SPDE 的模型都是最容易理解的随机场模型之一。为了支持这一说法，我们现在简要总结一下我们对基于 SPDE 的模型和相应计算方法的了解。在 5.1 小节中，我们介绍了一些已知的关于基于 SPDE 的模型的最重要的理论性质，在 5.2 小节中，我们介绍了有关相应近似方法的当前知识。</p>
<h3 id="5-1-SPDE-模型的性质">5.1  SPDE 模型的性质</h3>
<h4 id="5-1-1-具有常量参数的模型">5.1.1 具有常量参数的模型</h4>
<p>自 1960 年代初期（1950 年代的部分结果）以来，人们已经知道 D = \mathbb{R}^{d} 上的 SPDE (3) 的稳态解是具有 Matern 协方差矩阵函数的居中高斯场（Whittle，1954 年，1963 年）。因此，在这种情况下，解的理论性质可以从平稳高斯随机场的标准理论中获得（例如 Cram ´ er 和 Leadbetter，2004）。</p>
<p>由于算子 L 的特征值是根据拉普拉斯算子的特征值明确定义的，因此在球面等流形上考虑 (3) 的解的性质也很容易理解（例如，参见 Lang 和 Schwab，2015 年； Borovitskiy 等，2020 年）。特别是，在 D = \mathbb{R}^{d} 的情况下，指数 α 控制 H ̈ 较旧的解的连续性和可微性。</p>
<p>当在有界域 D ( \mathbb{R}^{d} 上考虑 SPDE (3) 时，必须向算子添加边界条件。通常，在实践中使用 Neumann 或 Dirichlet 边界条件。因此，解将是非平稳的，并且没有不再具有 Mat ́ ern 协方差矩阵。然而，Lindgren 等 (2011) 表明，对于 d = 1 和 Neumann 边界条件，解具有折叠的 Mat ́ ern 协方差矩阵  ̃κ,τ,α，这将类似于相应的域内部的 Mat ́ ern 协方差矩阵 κ,τ,α。因此，有人认为应该使用域 D，该域 D 扩展距离 δ，该距离至少是实际相关变程 ρ = 的两倍√ 8ν/κ 在感兴趣域 D0 之外。Khristenko 等 (2019) 进一步验证了此过程，他们将结果扩展到 d &gt; 1 以及当 D 是 \mathbb{R}^{d} 中的盒子时的 Neumann 和周期性边界条件。他们在特别表明最高范数 | ̃κ,τ,α − κ,τ,α|L\infty(D0) 可以用 o 来界定f δ/ρ 并且随着此项的增加，误差呈指数级渐近减小。因此，具有平稳参数的 SPDE 以及这种情况下边界条件的影响是很好理解的。</p>
<h4 id="5-1-2-非平稳-Whittle-Mat-泛化">5.1.2 非平稳 Whittle-Mat 泛化</h4>
<p>非平稳情况下的理论涉及更多；然而，在这种情况下，过程的特性也得到了很好的理解。考虑到凸多胞形 D \subset  \mathbb{R}^{d}, d \in  {1, 2, 3} 的广义 Whittle–Matern 域 (7)，我们从 Bolin 等那里知道。 (2020); Bolin 和 Kirchner (2020) 给出 α &gt; d/2（对应于 ν &gt; 0 在平稳情况下）的 SPDE 存在唯一解，κ 是一个本质上有界的函数，κ \in  L\infty(D)， H 是一个足够好的（Lipschitz 在 ̄ D 上连续且一致正定）矩阵值函数。赫尔曼等 (2020) 扩展了这一存在性结果，表明它在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span> 中的封闭、连通、可定向、光滑、紧凑的 2 曲面上考虑 SPDE (7) 时也成立，假设 κ、H 是光滑的，Harbrecht 等阿尔。 (2021) 在更一般的无边界流形上为模型得出了类似的结果</p>
<p>Cox 和 Kirchner (2020) 通过仅要求域 D 具有 Lipschitz 边界，并放宽对 H 的要求以仅假定本质有界性和一致正定性，进一步推广了 D \subset  \mathbb{R}^{d} 的情况。更重要的是，他们还表征了解 u 及其协方差矩阵函数的 Sobolev 和 H ̈ 旧正则性。因此，基于 SPDE 的模型的规律性在非平稳情况下也是已知的。</p>
<h4 id="5-1-3-归纳高斯测量和克里金法">5.1.3 归纳高斯测量和克里金法</h4>
<p>高斯场理论的关键工具之一是高斯测度的等价性和正交性。例如，这通常用于推导最大似然估计的一致性（Zhang，2004）。 Bolin 和 Kirchner（2020 年）展示了两个具有恒定参数的不同 SPDE 模型 (3) 何时生成等效测量的问题。这也表明，对于固定的 α 值，可以在填充渐近下一致地估计 τ，但不能估计 κ，这与高斯矩阵场的结果一致（Zhang，2004）。</p>
<p>对于统计应用程序，了解在模型中错误指定参数的影响也很重要。例如，Stein (1999) 在克里金法的背景下对稳态高斯随机场进行了彻底研究。对于非平稳广义 Whittle–Matern 场，类似的结果也具有普遍性。例如，Kirchner 和 Bolin (2021) 基于错误指定的参数 α、κ、τ 推导了球面上各向同性 Whittle–Matern 场线性预测的均匀渐近最优性条件。此外，Bolin 和 Kirchner (2021) 基于错误指定的参数 α、κ、H 推导了 \mathbb{R}^{d} 中有界域上广义 Whittle–Matern 场线性预测的均匀渐近最优性的条件。他们进一步推广了 Bolin 和 Kirchner 的结果（ 2020）通过推导两个广义 Whittle–Matern 场引起等效高斯测度的明确条件。据我们所知，广义 Whittle-Matern 是唯一一类已知此类理论结果的非平稳模型。</p>
<h3 id="5-2-近似的性质">5.2 近似的性质</h3>
<p>SPDE 模型的计算方法也很好理解。 Lindgren 等已经。 (2011) 结果表明，随着网格变得更精细，α \in  N 情况下的有限元近似分布收敛于精确解的分布。 Simpson 等扩展了这些结果。 (2016) 谁考虑了基于 SPDE 模型的对数高斯 Cox 过程 (3)</p>
<p>后来，一般的分数案例已经被彻底调查过了。这始于 Bolin 等的结果。 (2020) 导出了第 4.1 节中介绍的 sinc-Galerkin 近似 uh 的强误差 E(|u − uh|L2(D)) 的显式收敛率。博林等 (2018) 通过推导弱错误的显式收敛率扩展了这些结果 |E(g(u)) − E(g(uh))|对于足够平滑的泛函 g(\cdot )。 Cox 和 Kirchner (2020) 通过还为近似的协方差矩阵函数的误差提供显式收敛率，进一步扩展了结果。所有这些结果也适用于 Bolin 和 Kirchner（2020 年）的理性 SPDE 方法以及表面模型（Herrmann 等，2020 年）。</p>
<p>最近，SanzAlonso 和 Yang (2021a) 导出了包含在第 3.1 节中的简单分层模型以及二元分类模型中的 FEM 近似的后收缩率。这为如何根据所考虑的数据集的大小选择 FEM 基函数的数量提供了理论依据。</p>
<h2 id="6-应用">6 应用</h2>
<p>自 Lindgren 等发表以来。 (2011) 论文中，各种各样的应用程序利用了 R-INLA 包中可用的软件实现，以及使用了 SPDE 结构的专门实现。我们将重点介绍一些关键应用，这些应用展示了这些模型在具有实际复杂观测模型和大型分层随机场结构的应用问题中的实用性。</p>
<h3 id="6-1-使用时空-SPDE-进行疟疾建模">6.1 使用时空 SPDE 进行疟疾建模</h3>
<p>GMRF/SPDE 模型的第一个大规模应用是 Bhatt 等 (2015); Bertozzi-Villa 等 (2021)，它模拟了疟疾控制随时间的影响。结果表明，非洲的感染率在 2000 年至 2015 年间减半，估计有 5.42 亿至 7.63 亿（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>95</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">95\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">95%</span></span></span></span> 可信区间）避免的病例归因于预防性干预措施，例如经过杀虫剂处理的蚊帐</p>
<p>与医学数据一样，必须仔细考虑复杂的测量结构，使用时空 SPDE 模型来捕获其余模型组件无法自行处理的空间结构化效应。由于非洲足够大，任何地图投影都会引入变形，因此该模型直接建立在球形流形的子集上。尽管使用了空间静止模型，但通过消除由于任意地图投影引起的虚假非平稳性，结果可以根据地球上的测地线距离进行解释。该实现使用覆盖非洲的三角球形网格，以及来自第 4.3 节的时空 Kronecker 精度矩阵模型。</p>
<h3 id="6-2-EUSTACE-项目">6.2  EUSTACE 项目</h3>
<p>在分析过去的天气和气候时，一个挑战是合并来自多个数据源和测量类型的信息。近年来，卫星提供了大量数据，但测量数据与感兴趣的数量之间存在复杂的关系，包括空间和时间相关的噪声和卫星特定偏差。在全球某些地方，气象站数据的时间可以追溯到更早之前，但由于当地天气变化和仪器变化，这些数据在时间上具有持久性和不断变化的偏差。类似的挑战也适用于船舶的空气温度测量。 EUSTACE 大型合作项目（Rayner 等，2020 年）旨在重建全球每日时间尺度和 1/4 × 1/4 度空间分辨率的天气和气候。完整的问题，包括对自 1850 年以来所有 \sim 60, 000 天的每日最高和最低温度进行建模，大约有 1011 个值需要估计。</p>
<p>在高斯过程和克里金法的基本应用中，典型模型将单个随机场与几个协变量相结合，并将它们与具有高斯加性噪声的地理参考观测值联系起来。全球天气场具有在广泛的空间和时间尺度上运行的依赖结构，明确设计天气的协方差矩阵模型是不现实的。 EUSTACE 项目反而构建了一个分层模型，其中每个节点仅对完整行为的一个方面有贡献，例如缓慢变化的气候平均温度场、系统纬度效应和每日天气残差。联合地，这定义了关于连接多个时空图的图的马尔可夫随机场。正如在第 4.1 节中讨论的分数 SPDE 结构一样，得到的场总和不是马尔可夫随机场，但马尔可夫性质的计算优势仍然存在。</p>
<p>该项目通过与 R-INLA 软件中相同的非高斯观测近似技术，探索了处理每日最高和最低温度的非高斯行为的方法。为了使实现和计算时间易于管理，最终实现的方法不包括此，但该工作的各个方面可以在 Vandeskog 等中找到。 (2021b)。相反，实施了完全高斯方法，并使用迭代线性求解器计算所有潜在变量的条件分布，仅通过估计日平均温度将空间分辨率降低到 1.5 \cdot  1010，并将空间分辨率降低到 0.5 度。这些组件分为三类：</p>
<p>• 气候变化（\sim 3.5 \cdot  105 个节点）：2 个月 1 度季节性模式、5 年 5 度尺度气候变化、非线性纬度效应、高度效应、海岸效应和总体平均值 • 大尺度变化（\sim 1.8 \cdot  106 个节点）：3 个月 5 度场和气象站偏差随机效应 • 每日场（\sim 6 \cdot  104 × 2.5 \cdot  105 ≈ 1.5 \cdot  1010 个节点）：当地 0.5 度天气，卫星偏差场在2.5度分辨率</p>
<p>为了计算条件期望，使用了迭代条件方法，其中每个类别都根据其他类别有条件地求解，轮换直到收敛。日常类别将每一天视为条件独立的，这允许大规模并行计算，有 60 000 个单独的服务器任务。因此，最大的个体求解是针对大规模变化类别，计算图大小约为 200 万，使用精度矩阵的直接 Cholesky 分解</p>
<h3 id="6-3-神经影像学">6.3 神经影像学</h3>
<p>在流形上制定类似 Matern 的随机场的能力为应用到基于协方差矩阵的模型非常难以接近的领域开辟了道路。一种这样的应用是神经成像。具体而言，功能磁共振成像 (fMRI) 是一种流行的神经成像技术，用于定位由任务或刺激激活的大脑区域。传统的体积 fMRI 数据包括对大脑中数千个三维体积测量的时间序列的观测。分析 fMRI 数据的常用方法是通过经典的一般线性建模 (GLM) 方法（Friston 等，1994）。</p>
<p>GLM 方法没有任何明确的空间依赖性统计模型，但通过数据的预平滑和多个假设检验的后校正来隐含地考虑空间依赖性，以寻找激活区域，这被认为是有问题的（见，例如 Eklund 等，2016 年）。然而，很少对 fMRI 中的空间依赖性进行显式建模，这主要是因为空间模型的计算成本很高。这个问题最近通过应用 SPDE 方法来定义具有空间先验的全脑 fMRI 分析方法而得到解决（Sid en 等，2021 年）</p>
<p>尽管它很受欢迎，但传统的 fMRI 分析有一些局限性，例如数据包括来自许多不同组织类型的观测结果，而众所周知，神经元活动只发生在灰质中。皮质表面 fMRI (cs-fMRI) 通过将皮质灰质表示为二维流形表面来解决这个问题 (Fischl, 2012)。该方法最近越来越受欢迎，因为它还提供了更好的可视化、降维和改进了受试者皮质区域的对齐。此外，流形表示允许对位置之间的距离进行更具神经生物学意义的测量，这对于分析非常重要。从统计角度分析此类数据的主要困难是需要对皮质表面的空间依赖性进行建模。最近通过使用 SPDE 方法定义 cs-fMRI 数据的第一个贝叶斯 GLM 方法（Mejia 等，2020b）克服了这一困难，与忽略空间依赖性的标准 GLM 方法相比，该方法显示出高度准确（Spencer 等）等，2021 年）。在图 1 中，我们展示了对患者皮质表面的广义 Whittle–Matern 场的模拟。</p>
<p>上面提到的模型可以有效地发现执行特定任务时大脑的哪些区域是活跃的。最近备受关注的另一种研究大脑的方法是采用更全面的方法，在没有特定刺激的情况下观测大脑的功能组织。此任务的常用方法是独立成分分析 (ICA)，它在观测患者组时可靠地工作 (Calhoun 等, 2001)。在主题级别执行估计更具挑战性，并且在主题级别考虑空间依赖性对于减少噪声变得更加重要。这对于 fMRI 和 cs-fMRI 数据来说也很难做到，但 SPDE 方法最近也促进了空间 ICA 方法的开发，用于估计功能性大脑网络（Mejia 等，2020a）。</p>
<p>类似的技术也已应用于其他解剖流形，例如心脏。为了从电图模拟局部激活时间，Coveney 等 (2020) 估计了多种人类心房的概率激活时间。三角流形网格是根据测量的个体估计的，然后进行平滑处理。然后将电极位置投影到最近的网格点，并使用 R-INLA 软件将第 4.3 节中的 Kronecker 乘积精度矩阵模型用于流形上的时空激活过程。</p>
<h3 id="6-4-地震学和材料科学">6.4 地震学和材料科学</h3>
<p>虽然 SPDE 模型在空间统计中最常见的应用是 2 维空间，有时会增加时间，但在地震学中，感兴趣的数据和问题通常是 3 维的。希尔伯特空间构造中使用的有限元方法对四面体化的工作方式与对三角剖分的工作方式相同。这被张等利用了。 (2016)，估计美国西部 700 公里深度的地震速度。在这个复杂的反演问题中，他们将 SPDE 模型应用于地下速度场以及地表的地震源和接收器场。他们还提供了四面体网格的有限元矩阵 C 和 G 的几何推导，此后已在 <a target="_blank" rel="noopener" href="https://github.com/finnlindgren/inlamesh3d">https://github.com/finnlindgren/inlamesh3d</a> 的实验性 R 包中实现。</p>
<p>多孔材料的统计建模是另一种应用，在非常不同的空间尺度上，它也需要 3 维空间中的模型。 Barman 和 Bolin（2018 年）使用 SPDE 方法设计了一个模型，用于研究多孔乙基纤维素/羟丙基纤维素聚合物混合物，该混合物用作控制药片药物释放的涂层。该模型后来被用于研究孔隙几何形状如何影响通过聚合物薄膜的扩散传输（Barman 等，2019 年）。</p>
<h3 id="6-5-生态学中的点过程">6.5 生态学中的点过程</h3>
<p>在 SPDE 构造中使用局部基函数的好处之一是与生态学和流行病学中常见的基于图的马尔可夫随机场模型非常相似。这使得从区域聚合计数到连续域点过程模型的步骤非常小。 Simpson 等详细分析了非齐次泊松过程可能性的直接近似。 (2016)，此后作为 inlabru 软件的一部分实现了自动化（Bachl 等，2019 年；Yuan 等，2017 年）。</p>
<p>当 η(\mathbf{s}) 是空间评估的线性预测变量表达式时，非齐次泊松点过程 Y = {y1,\ldots , yM}, y_i \in  D, 在域 D 上具有强度 λ(\mathbf{s}) = \exp{η(\mathbf{s})} 具有对数似然</p>
<p>l(η; Y) = − ∫ D \exp{η(\mathbf{s})} dD(\mathbf{s}) + M sum  i=1 η(y_i)</p>
<p>其中积分是流形域 D 的表面积分。积分通常是难以处理的，但是当 η(\mathbf{s}) 是从第 2.7 节中的局部基函数构建时，有几个选项可用。例如，离散化 sum N j=1 w j \exp{η(s j)}，其中 s j 是网格节点，w j = \langle\psi_j, 1\rangle 提供了良好且稳定的近似。生成的似然表达式不是 η 的泊松模型似然，因为这两个和涉及不同的空间位置。然而，它非常相似，很容易在 R-INLA 软件中实施，并使其自动化，就像在 inlabru 中一样。</p>
<p>inlabru 软件的一个关键动机是允许更轻松地指定此类模型以及更复杂的模型，例如横断面调查的距离采样，其中点过程仅沿线观测，例如从一艘穿越海洋的船上。然后表面积分被线积分有效地代替，并且检测到一个点（通常是一个动物或一组动物）的概率也可以被合并。然而，此类模型不一定会产生关于所产生的细化泊松点过程强度参数的对数线性表达式，λ(\mathbf{s})P（在 s 处检测到的点 | 点存在于 s 处）。为了解决这个问题，inlabru 方法迭代地线性化给定的预测表达式，并对线性化版本应用 INLA 方法，直到找到后验模式。</p>
<p>R 包 dsm（Miller 等，2021 年）中提供了距离采样的频率论方法，该方法使用与本征平稳随机场模型密切相关的惩罚样条平滑器。</p>
<h2 id="7-相关方法">7 相关方法</h2>
<p>在这里，我们重点介绍一些相关或对比的技术和方法。进一步的联系、相关和对比方法可以在 Cressie 和 Wikle (2011) 以及 Wikle 等中找到。 (2019)，涵盖分层模型设置中的时空模型，以及各种计算方法，包括模型评估。后者正成为一个越来越重要的话题，因为后验均值（或频率点估计）之间的均方误差等传统基本度量仅提供有限的见解。为了有意义地比较复杂的空间和时空估计技术，必须使用将预测的估计不确定性考虑在内的比较分数，例如对数密度和对数概率分数（分别用于连续和离散结果)、CRPS（连续排名概率得分）或 Dawid-Sebastiani（来自高斯分布的对数密度得分，对于其他分布的期望和方差也是一个合适的得分），参见 Gneiting 和 Raftery (2007) .对于 SPDE 方法生成的稀疏精度矩阵，也可以应用具有此类分数的留一法交叉验证（Vehtari 等，2017 年）（Ferkingstad 等，2017 年）。</p>
<h3 id="7-1-过程先验与平滑惩罚">7.1 过程先验与平滑惩罚</h3>
<p>正如 Miller 等最近所讨论的那样，再现核 Hilbert 空间的理论提供了频繁惩罚样条估计器和贝叶斯高斯过程先验之间的密切联系。 （2020）。在开发随机 PDE 方法的同时，基于 PDE 的惩罚的相关开发也在进行，以解决类似的问题，例如复杂形状域和非欧几里得流形（Sangalli 等，2013 年；Sangalli，2021 年）。</p>
<p>正如 2.2 节中提到的，惩罚最小化器（这里通常与条件期望相同或相似）和完全随机过程之间的主要区别在于，惩罚最小化器与随机过程具有相同的 RKHS，从根本上比随机过程更平滑过程实现。这也体现在对于高维高斯分布（其中随机场处于极端无限极限），大部分概率质量远离分布的期望，并且在量化预测不确定性时这种偏差是必不可少的。惩罚方法提供的点估计量或条件期望的方差不应与过程的完整后验分布提供的预测不确定性相混淆。</p>
<h3 id="7-2-谱模型构造和广义-Whittle-Matern-场">7.2 谱模型构造和广义 Whittle-Matern 场</h3>
<p>使用 Whittle SPDE 的一个重要方面是将 Matern 协方差矩阵族推广到流形上的过程，同时保持局部几何解释。 Matern 协方差矩阵模型对平滑流形的 SPDE 泛化保留了原始模型的所有可微性和马尔可夫子集性质，并且对于短程渐近等价。谱表示与拉普拉斯算子及其流形版本的特征函数和特征值相关联。 Lang 和 Pereira（2021 年）以及 Harbrecht 等最近使用高阶数值方法对黎曼流形上的偏微分方程扩展了 4.1 节中讨论的分数算子方法。 (2021)，涉及多项式和小波基展开。在 Porcu 等中可以找到在球面上构建有效模型的其他方法。 (2016)。</p>
<p>在 \mathbb{R}^{d} 上，调和 \exp(is\cdot k) 的拉普拉斯算子的特征值为 −|k|2，k \in  \mathbb{R}^{d}，在球面上，\lambda_k = −k(k + 1) 是球谐 Yk 的特征值， k 阶的 m \in  {0, 1, 2,\ldots }, m \in  {−k,\ldots , k}。在文献中，谱结构通常用于定义新的模型族，例如用于时空模型（Stein，2005）。在使用 Whittle SPDE 表示将 Matern 模型推广到球面和其他非欧几里得流形时，拉普拉斯算子的谱表示起着关键作用。相比之下，Guinness 和 Fuentes（2016 年）以 Legendre-Matern 的名义引入的球形协方差矩阵在频谱定义中使用 k2 而不是 Whittle SPDE 泛化中出现的 k(k + 1)。此外，构造没有考虑对模型本身的频谱表示的影响，导致不同的算子功率，并且在频谱到协方差矩阵转换中缺少第 2.5 节中的 2k + 1 因子。因此，在 Whittle-Mat ́ ern 泛化具有 {\kappa^{2} + k(k + 1)}α 的地方，Legendre-Mat ́ ern 版本具有 (2k + 1)(\kappa^{2} + k2)α−1/2，这不能使用拉普拉斯算子的特征值很容易重新表述。这些问题使得 Matern 模型的替代泛化变得不那么自然，因为它失去了泛化 Whittle 表示的所有马尔可夫连接，以及精度算子的简单形式，这不能轻易地通过拉普拉斯算子的幂来表达。当仅考虑正定性的理论上有效的表达式时，很容易错过这些影响，并且在几何局部可解释的微分算子中建立更复杂的模型构造可能会提供对理论的更好的直观理解</p>
<h3 id="7-3-全局基函数">7.3 全局基函数</h3>
<p>在第 2.7 节中，使用局部支持的基函数来生成稀疏精度矩阵并在对测量进行调节时保留这种稀疏性。在另一个极端，可以使用全局基函数，这样选择使得基重的模型精度矩阵是对角线的，但后验精度矩阵通常是密集的。 Karhunen-Lo` eve (K-L) 展开式使用协方差矩阵算子或等效的精度算子的特征函数。这种类型的有限维谱表示的好处是，它可以为较少数量的基函数生成与真实协方差矩阵的近似值。主要缺点是评估特征函数的计算成本，因为这些取决于模型参数。另一种选择是使用拉普拉斯算子的本征函数，这些函数涉及类傅立叶谱表示。这些仅取决于域的形状，并且可以在分析开始时进行数值计算（参见 Solin 和 S ̈ arkk ̈ a, 2020，其中一种方法）。然而，由于可能需要高频来提供对真实模型的良好近似，因此这种方法对于一般领域来说仍然很昂贵。此外，典型的数值解法是使用与用于计算整个条件期望相同的有限元方法来求解特征值。因此，仅当可以比稀疏精度矩阵本身更有效地使用特征函数时，以数值方式计算特征函数才有用。对于 K-L 和谐波基函数，希尔伯特空间内积产生对角先验精度矩阵。由于 (11) 中 A&gt;\mathbf{Q}_eA 项的结构，当数据结构由于生成的密集精度矩阵而不允许有效的后验分布计算时，就会出现实际应用中的主要问题。这意味着谐波基函数最适用于只需要几个频率的非常平滑的场，或者可以进行快速傅立叶变换的场。</p>
<p>在球面上使用球谐函数时，在选择勒让德多项式实现时必须小心，因为某些实现在阶数高于 \sim 40 时在数值上不稳定。例如，首先显式构造多项式系数，然后评估它们的方法，分解对于 \sim 40 以上的订单，包括 pracma 和 orthopolynom 包中的实现（Borchers，2021；Novomestky，2013）。由于这些不稳定的实现，地球上的空间分辨率被限制在大约 360/40 = 9 度或 \sim 1, 000 公里的波长，使其不适合精细分辨率问题，例如第 6.2 节中讨论的 EUSTACE 项目。然而，GSL 库实现（Galassi，2018 年；Hankin，2006 年）对于更高的阶次似乎是稳定的。</p>
<p>一种数值替代方法是通过广义特征值问题 GV = CVΛ 计算三角剖分上的拉普拉斯特征函数。这非常接近连续域拉普拉斯算子的特征函数，并且也适用于球面以外的其他流形。这将是对 Lee 和 Haran (2021) 使用的直接基于图的特征函数的轻微修改</p>
<h3 id="7-4-其他精度矩阵近似方法">7.4 其他精度矩阵近似方法</h3>
<p>空间过程的每一种计算方法都可以（至少）以某种方式来看待；作为给定模型的近似值，或作为其自身专门构建的模型。在文献中，这两种观点都被使用，但由于近似的结构可能非常特定于构造的细节，因此第二种方法无法提供对两种方法之间异同的有用见解。相反，我们发现采用第一种方法更有用，并考虑方法的连续域可解释性。</p>
<p>表示 SPDE 解的有限元希尔伯特空间方法产生的马尔可夫模型与其他几种构造给定连续域模型的计算有效近似的方法有着密切的联系。</p>
<p>正如 Lindgren 等开发 SPDE/GMRF 链接的动机之一。 （2011）是为了弥合离散马尔可夫模型和连续协方差矩阵模型之间的差距，有限元三角剖分图和其他图方法之间有进一步的联系。最近的两个例子是 Sanz-Alonso 和 Yang (2021b) 以及 Dunson 等 (2021)，在同一种图上构造不同的局部拉普拉斯近似。这两篇论文中的第一篇还展示了这些方法之间的广泛联系，以及有限元结构如何在可用的设置中提供对连续域模型更好的近似</p>
<p>另一种方法是 Datta 等(2016)的最近邻高斯过程 (NNGP) 构造。 ，它采用给定的协方差矩阵函数，本质上，为给定的、有序的空间位置序列构造精度矩阵的不完整 Cholesky 分解。通过计算给定先前包含的点的精确条件分布，获得离散化模型。好处是，不是包括出现在马尔可夫精度矩阵的 Cholesky 分解中的 Cholesky 填充，而是只包括以前的邻居。缺点是生成的表示取决于包含空间位置的顺序，这可能导致源协方差矩阵和 NNGP 协方差矩阵之间存在较大差异。相反，对于完整的 Cholesky 分解，节点的排序只影响计算分解的稀疏性（Rue 和 Held，2005）。然而，由于 NNGP 构建的相对速度，潜在的扩展可能是使用 NNGP 模型作为大规模迭代求解器方法中的快速预处理器。这也适用于块状结构，例如 Katzfuss (2017)；基罗斯等 (2021);佩鲁齐等 （2020）。</p>
<h2 id="8-讨论">8 讨论</h2>
<p>自 Lindgren 等发表 10 年后。 (2011)，用于构建空间和时空高斯随机场的计算高效表示的随机 PDE 方法已在广泛的实际应用中证明了其价值。现在空间统计的计算方法多种多样，但许多缺乏与所需连续域性质相关的保真度的强有力的理论保证，或者仅限于特定应用。相比之下，SPDE 方法利用了高斯过程的不同表示及其依赖结构之间的紧密联系，以及模型构建本身与计算方法的相对分离，提供了强有力的理论保证和高效的实现。这使得高斯随机场模型可以被纳入各种层次模型中，并且该方法继续扩展到新的应用领域和更通用的依赖结构。</p>
<p>在通用性和计算效率之间总是存在权衡。 SPDE 方法的优势在于，可以从已经能够求解大型系统的确定性 PDE 求解器的经过充分研究的方法中改编方法。部分挑战在于，随机场的典型时空精度算子的阶数远高于纯物理驱动的 PDE 中常见的典型普通拉普拉斯算子，这使得迭代求解器的标准预处理方法效率低下。然而，通过利用局部块结构以及多重网格方法和并行求解器，这些方法显示出比当前直接 Cholesky 方法可以解决的更大问题的前景，这些问题可用于多达几百万个元素； EUSTACE 项目中最大的单场求解有近 200 万个节点。</p>
<p>当前活跃的研究领域包括更一般的时空扩展、非平稳性和各向异性模型及其组合。我们预计这将有助于将 SPDE 模型的实际用途扩展到复杂时空数据的现实模型。</p>
<p>虽然与这些方法相关的实际实施和预处理成本仍然是事实，并且所有高级空间和时空建模技术都是如此，但 Lindgren 等的结束语。 (2011) 从软件用户的角度来看，“当需要高效计算时，这种成本是不可避免的”现在不再像 2011 年那样正确。 inlabru等接口能够将所有的内部记账和线性代数封装在内部代码中，让软件用户更专注于构建结构化和几何可解释的模型。</p>
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class="title">Vecchia 近似似然法</div></div></a></div><div><a href="/posts/fe6491e.html" title="Vecchia 近似似然法的通用框架"><img class="cover" src="/img/coffe_05.png" alt="cover"><div class="content is-center"><div class="date"><i class="far fa-calendar-alt fa-fw"></i> 2023-01-30</div><div class="title">Vecchia 近似似然法的通用框架</div></div></a></div></div></div></div><div class="aside-content" id="aside-content"><div class="sticky_layout"><div class="card-widget" id="card-toc"><div class="item-headline"><i class="fas fa-stream"></i><span>目录</span><span class="toc-percentage"></span></div><div class="toc-content"><ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#1-%E7%AE%80%E4%BB%8B"><span class="toc-text">1 简介</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#1-1-%E5%8D%8F%E6%96%B9%E5%B7%AE%E7%9F%A9%E9%98%B5%E8%BF%98%E6%98%AF%E7%B2%BE%E5%BA%A6%E7%9F%A9%E9%98%B5%EF%BC%9F"><span class="toc-text">1.1 协方差矩阵还是精度矩阵？</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#1-2-%E6%9C%80%E8%BF%91%E7%9A%84%E4%B8%80%E4%BA%9B%E5%BA%94%E7%94%A8"><span class="toc-text">1.2 最近的一些应用</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#1-3-%E6%9C%AC%E6%96%87%E5%AE%89%E6%8E%92"><span class="toc-text">1.3 本文安排</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#2-%E4%B8%BB%E8%A6%81%E6%80%9D%E6%83%B3%E6%A6%82%E8%BF%B0"><span class="toc-text">2 主要思想概述</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#2-1-%E5%8D%8F%E6%96%B9%E5%B7%AE%E7%9F%A9%E9%98%B5%E5%92%8C%E9%9A%8F%E6%9C%BA%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B"><span class="toc-text">2.1 协方差矩阵和随机偏微分方程</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#2-2-%E7%B2%BE%E5%BA%A6%E7%AE%97%E5%AD%90%E5%92%8C%E5%86%8D%E7%94%9F%E6%A0%B8-Hilbert-%E7%A9%BA%E9%97%B4"><span class="toc-text">2.2 精度算子和再生核 Hilbert 空间</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#2-3-%E5%94%AF%E4%B8%80%E6%80%A7%E5%92%8C%E6%9C%AC%E5%BE%81%E5%B9%B3%E7%A8%B3%E9%9A%8F%E6%9C%BA%E5%9C%BA"><span class="toc-text">2.3 唯一性和本征平稳随机场</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#2-4-%E8%B0%B1%E8%A1%A8%E7%A4%BA"><span class="toc-text">2.4 谱表示</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#2-5-%E6%B5%81%E5%BD%A2"><span class="toc-text">2.5 流形</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#2-6-%E9%9D%9E%E5%B9%B3%E7%A8%B3%E6%A8%A1%E5%9E%8B"><span class="toc-text">2.6 非平稳模型</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#2-7-%E5%B1%80%E9%83%A8%E6%94%AF%E6%8C%81%E7%9A%84%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9%E7%A9%BA%E9%97%B4%E5%9F%BA%E5%92%8C%E7%A6%BB%E6%95%A3%E7%B2%BE%E5%BA%A6"><span class="toc-text">2.7 局部支持的希尔伯特空间基和离散精度</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#3-%E5%AE%9E%E7%94%A8%E7%9A%84%E7%A9%BA%E9%97%B4%E4%BC%B0%E8%AE%A1%E5%92%8C%E6%8E%A8%E6%96%AD"><span class="toc-text">3 实用的空间估计和推断</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#3-1-%E5%90%AB%E5%99%AA%E5%A3%B0%E8%A7%82%E6%B5%8B%E4%B8%8B%E7%9A%84%E6%9D%A1%E4%BB%B6%E5%88%86%E5%B8%83"><span class="toc-text">3.1 含噪声观测下的条件分布</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#3-2-%E6%B7%BB%E5%8A%A0%E6%A8%A1%E5%9E%8B%E7%BB%84%E4%BB%B6"><span class="toc-text">3.2 添加模型组件</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#3-3-%E8%B4%9D%E5%8F%B6%E6%96%AF%E6%8E%A8%E6%96%AD%E5%92%8C%E9%9D%9E%E9%AB%98%E6%96%AF%E8%A7%82%E6%B5%8B"><span class="toc-text">3.3 贝叶斯推断和非高斯观测</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#4-%E9%87%8D%E8%A6%81%E6%89%A9%E5%B1%95"><span class="toc-text">4 重要扩展</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#4-1-%E4%B8%80%E8%88%AC%E5%88%86%E6%95%B0%E5%B9%82%E7%9A%84%E6%96%B9%E6%B3%95"><span class="toc-text">4.1 一般分数幂的方法</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#4-2-%E9%9D%9E%E9%AB%98%E6%96%AF%E6%A8%A1%E5%9E%8B"><span class="toc-text">4.2 非高斯模型</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#4-3-%E6%97%B6%E7%A9%BA%E8%BF%87%E7%A8%8B"><span class="toc-text">4.3 时空过程</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#5-%E7%90%86%E8%AE%BA%E4%BF%9D%E8%AF%81"><span class="toc-text">5 理论保证</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#5-1-SPDE-%E6%A8%A1%E5%9E%8B%E7%9A%84%E6%80%A7%E8%B4%A8"><span class="toc-text">5.1  SPDE 模型的性质</span></a><ol class="toc-child"><li class="toc-item toc-level-4"><a class="toc-link" href="#5-1-1-%E5%85%B7%E6%9C%89%E5%B8%B8%E9%87%8F%E5%8F%82%E6%95%B0%E7%9A%84%E6%A8%A1%E5%9E%8B"><span class="toc-text">5.1.1 具有常量参数的模型</span></a></li><li class="toc-item toc-level-4"><a class="toc-link" href="#5-1-2-%E9%9D%9E%E5%B9%B3%E7%A8%B3-Whittle-Mat-%E6%B3%9B%E5%8C%96"><span class="toc-text">5.1.2 非平稳 Whittle-Mat 泛化</span></a></li><li class="toc-item toc-level-4"><a class="toc-link" href="#5-1-3-%E5%BD%92%E7%BA%B3%E9%AB%98%E6%96%AF%E6%B5%8B%E9%87%8F%E5%92%8C%E5%85%8B%E9%87%8C%E9%87%91%E6%B3%95"><span class="toc-text">5.1.3 归纳高斯测量和克里金法</span></a></li></ol></li><li class="toc-item toc-level-3"><a class="toc-link" href="#5-2-%E8%BF%91%E4%BC%BC%E7%9A%84%E6%80%A7%E8%B4%A8"><span class="toc-text">5.2 近似的性质</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#6-%E5%BA%94%E7%94%A8"><span class="toc-text">6 应用</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#6-1-%E4%BD%BF%E7%94%A8%E6%97%B6%E7%A9%BA-SPDE-%E8%BF%9B%E8%A1%8C%E7%96%9F%E7%96%BE%E5%BB%BA%E6%A8%A1"><span class="toc-text">6.1 使用时空 SPDE 进行疟疾建模</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#6-2-EUSTACE-%E9%A1%B9%E7%9B%AE"><span class="toc-text">6.2  EUSTACE 项目</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#6-3-%E7%A5%9E%E7%BB%8F%E5%BD%B1%E5%83%8F%E5%AD%A6"><span class="toc-text">6.3 神经影像学</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#6-4-%E5%9C%B0%E9%9C%87%E5%AD%A6%E5%92%8C%E6%9D%90%E6%96%99%E7%A7%91%E5%AD%A6"><span class="toc-text">6.4 地震学和材料科学</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#6-5-%E7%94%9F%E6%80%81%E5%AD%A6%E4%B8%AD%E7%9A%84%E7%82%B9%E8%BF%87%E7%A8%8B"><span class="toc-text">6.5 生态学中的点过程</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#7-%E7%9B%B8%E5%85%B3%E6%96%B9%E6%B3%95"><span class="toc-text">7 相关方法</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#7-1-%E8%BF%87%E7%A8%8B%E5%85%88%E9%AA%8C%E4%B8%8E%E5%B9%B3%E6%BB%91%E6%83%A9%E7%BD%9A"><span class="toc-text">7.1 过程先验与平滑惩罚</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#7-2-%E8%B0%B1%E6%A8%A1%E5%9E%8B%E6%9E%84%E9%80%A0%E5%92%8C%E5%B9%BF%E4%B9%89-Whittle-Matern-%E5%9C%BA"><span class="toc-text">7.2 谱模型构造和广义 Whittle-Matern 场</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#7-3-%E5%85%A8%E5%B1%80%E5%9F%BA%E5%87%BD%E6%95%B0"><span class="toc-text">7.3 全局基函数</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#7-4-%E5%85%B6%E4%BB%96%E7%B2%BE%E5%BA%A6%E7%9F%A9%E9%98%B5%E8%BF%91%E4%BC%BC%E6%96%B9%E6%B3%95"><span class="toc-text">7.4 其他精度矩阵近似方法</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#8-%E8%AE%A8%E8%AE%BA"><span class="toc-text">8 讨论</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE"><span class="toc-text">参考文献</span></a></li></ol></div></div></div></div></main><footer id="footer"><div 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